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Synthetic division

In Mathematics, there are two different methods to divide the polynomials. One is the long division method. Another one is the synthetic division method. Among these two methods, the shortcut method to divide polynomials is the synthetic division method. It is also called the polynomial division method of a special case when it is dividing by the linear factor.  It replaces the long division method .  In certain situations, you can find this method easier. In this article, we will discuss what the synthetic division method is, how to perform this method, steps with more solved examples.

Table of Contents:

  • How to Perform Synthetic Division
  • Advantages and Disadvantages
  • Practice Questions

Synthetic Division of Polynomials

The Synthetic division is a shortcut way of polynomial division, especially if we need to divide it by a linear factor. It is generally used to find out the zeroes or roots of polynomials and not for the division of factors. Thus, the formal definition of synthetic division is given as:

“Synthetic division can be defined as a simplified way of dividing a polynomial with another polynomial equation of degree 1 and is generally used to find the zeroes of polynomials”

This division method is performed manually with less effort of calculation than the long division method. Usually, a binomial term is used as a divisor in this method, such as x – b .

If we divide a polynomial P(x) by a linear factor (x-a), which of the polynomial of the degree 1, Q(x) is quotient polynomial and R is the remainder, which is a constant term. We use the synthetic division method in the context of the evaluation of the polynomial using the remainder theorem, wherein we evaluate the polynomial P(x) at “a” while dividing the polynomial P(x) by the linear factor. (i.e) P(x)/(x-a).

Mathematically, it can be represented as follows:

P(x)/Q(x) = P(x)/(x-a) = Quotient + [Remainder/(x-a)]

P(x)/(x-a) = Q(x) +[R/(x-a)]

Hence, we can use the synthetic division method to find the remainder quickly, if “a” is the factor of the polynomial.

In the synthetic division method, we use only the numbers for calculation and this method avoids the usage of the variables.

  • We can perform the synthetic division method, only if the divisor is a linear factor.
  • In the synthetic division method, we will perform multiplication and addition, in the place of division and subtraction, which is used in the long division method.

How to Perform a Synthetic Division?

If we want to divide polynomials using synthetic division, you should be dividing it by a linear expression and the first number or the leading coefficient should be a 1. This division by linear denominator is also called division through Ruffini’s rule(paper-and-pencil computation).

The requirements to perform the synthetic process method is given below:

  • The divisor of the given polynomial should be of degree 1. It means that the exponent of the given variable should be 1. Such kind of divisor is considered as the linear factor.
  • The coefficient of the divisor variable (say x) should be also equal to 1.

The process of the synthetic division will get messed up if the divisor of the leading coefficient is other than one. In case if the leading coefficient of the divisor is other than 1 while performing the synthetic division method, solve the problem carefully.

The basic Mantra to perform the synthetic division process is”

“Bring down, Multiply and add, multiply and add, Multiply and add, ….”

For example , we can use the synthetic division method to divide a polynomial of 2 degrees by x + a or x – a, but you cannot use this method to divide by x 2 + 3 or 5x 2 – x + 7.

If the leading coefficient is not 1, then we need to divide by the leading coefficient to turn the leading coefficient into 1. For example, 4x – 1 would become x – ¼ and 4x+9 would become x + 9/4. If the synthetic division is not working, then we need to use long division.

Steps for Polynomial Synthetic Division Method

Following are the steps required for Synthetic Division of a Polynomial:

Advantages and Disadvantages of Synthetic Division Method

The advantages of using the synthetic division method are:

  • It requires only a few calculation steps
  • The calculation can be performed without variables
  • Unlike the polynomial long division method, this method is a less error-prone method

The only disadvantage of the synthetic division method is that this method is only applicable if the divisor of the polynomial expression is a linear factor.

Synthetic Division Examples

Following the steps as per explained above, to divide the polynomials given. Thus, we can get;

Synthetic Division Example 1

Synthetic Division Example 1

Example 2: 

As per the given question; we have two polynomials in numerator and denominator. The denominator consists of a linear equation, so we can easily apply the synthetic division method here.

Follow the step by step method as given below:

how to solve a synthetic division problem

Following the same steps as per previous examples.

Synthetic Division Example 3

Synthetic Division Example 3

As we know, the step to solve the given equation by synthetic division method, we can write;

Synthetic Division Example 4

Now, divide the answer obtained in step 6 by 2. Hence, we get

Solving the given expression, by step by step method, we get;

Synthetic Division Example 5

Synthetic Division of polynomials Practice Questions

Solve the following problems:

  • Find the quotient and remainder of the polynomial 2x 3 -7x 2 +0x+11, when it is divided by a linear factor x-3.
  • Solve the following polynomial equation and find its quotient and remainder.  (9a 2 -39a-30)/(a-5)
  • Find Q(x) and R for the polynomial, P(x)=m 3 -3m+4 divided by the linear factor m-1.

Frequently Asked Questions on Synthetic Division

What is meant by synthetic division.

The synthetic division method is a special method of dividing polynomials. This method is a special case of dividing a polynomial expression by a linear factor, in which the leading coefficient should be equal to 1.

What are the requirements of the synthetic division method?

The requirements of the synthetic division method are: The divisor of the polynomial expression must have a degree of one (linear factor) The leading coefficient of the variable in the divisor should be equal to 1.

What is the Main Use of Synthetic Division?

Synthetic division is mainly used to find the zeroes of roots of polynomials.

When Can You Use Synthetic Division?

Synthetic division is used when a polynomial is to be divided by a linear expression and the leading coefficient (first number) must be a 1. For example, any polynomial equation of any degree can be divided by x + 1 but not by x 2 +1

Why is Synthetic Division Important?

Synthetic division is useful to divide polynomials in an easy and simple way as it breaks down complex equations into smaller and easier equations.

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Synthetic Division

In algebra, the synthetic division is one of the methods used to manually perform the Euclidean division of polynomials. The division of polynomials can also be done using the long division method. But, in comparison to the long division method of polynomials, the synthetic division requires lesser writing and fewer calculations. That means the synthetic division is the shorter method of the traditional long-division of a polynomial for the special cases when dividing by a linear factor.

Let us understand the method to perform the synthetic division of polynomials in detail using solved examples.

What is Synthetic Division?

Synthetic division is a method used to perform the division operation on polynomials when the divisor is a linear factor. One of the advantages of using this method over the traditional long method is that the synthetic division allows one to calculate without writing variables while performing the polynomial division , which also makes it an easier method in comparison to the long division .

We can represent the division of two polynomials in the form: p(x)/q(x) = Q(x) + R/(q(x))

  • p(x) is the dividend
  • q(x) is the linear divisor
  • Q(x) is quotient
  • R is remainder

Synthetic Division of Polynomials Definition

When we divide a polynomial p(x) by a linear factor (x - a) (which is a polynomial of degree 1), Q(x) is the quotient polynomial and R is the remainder .

p(x)/q(x) = p(x)/(x- a) = Quotient + (Remainder/(x - a))

p(x)/(x - a) = Q(x) + (R/(x - a))

The coefficients of p(x) are taken and divided by the zero of the linear factor.

We use synthetic division in the context of the evaluation of the polynomials by the remainder theorem, wherein we evaluate the value of p(x) at "a" while dividing (p(x)/(x - a)). That is, to find if "a" is the factor of the polynomial p(x), use the synthetic division to find the remainder quickly. Let us understand this better using the example given below.

Synthetic Division Example

Richard sells apples. The previous day, his profits were x, and today, his profits are ((x × x) - 2). If the number of apples he sold was (x + 2), what was the profit made per apple? We obtain the solution by modelling the equation as (x 2 + x - 2) ÷ (x + 2).

Step 1: Write the coefficients of the dividend inside the box and zero of x + 2 as the divisor.

Step 2: Bring down the leading coefficient 1 to the bottom row.

Step 3: Multiply -2 by 1 and write the product -2 in the middle row.

Step 4: Add 1 and -2 in the second column and write the sum -1 in the bottom row.

Step 5: Now, multiply -2 by -1 (obtained in step 4) and write product 2 below -2.

Step 6: Add -2 and 2 in the third column and write the sum 0 in the bottom row.

Step 7: The bottom row gives the coefficient of the quotient. The degree of the quotient is one less than that of the dividend. So, the final answer is x - 1 + 0/(x + 2) = x - 1.

Please note that the last box in the bottom row gives the remainder.

Synthetic division of polynomials example

The profit per apple is given by (x - 1).

Synthetic Division vs Long Division

Let us see how long division differs from the synthetic division of polynomials by comparing both methods. In the example given below, we will perform the division of the polynomial 4x 2 - 5x - 21 by a linear polynomial x - 3.

dividing polynomial by binomial

In the example given below, another polynomial 2x 2 + 3x - 1 is divided by a linear polynomial x + 1. When a polynomial P(x) is to be divided by a linear factor, we write the coefficients alone, bring down the first coefficient, multiply, and add. Repeat the multiplication and addition until we reach the end term of the polynomial.

synthetic division of polynomials

Using synthetic division, we can perform complex division and obtain the solutions easily.

Synthetic Division Method

The following are the steps while performing synthetic division and finding the quotient and the remainder. We will take the following expression as a reference to understand it better: (2x 3 - 3x 2 + 4x + 5)/(x + 2)

  • Check whether the polynomial is in the standard form .
  • Write the coefficients in the dividend's place and write the zero of the linear factor in the divisor's place.
  • Bring the first coefficient down.

synthetic division - step 3

  • Multiply it with the divisor and write it below the next coefficient.
  • Add them and write the value below.

synthetic division - step 5

  • Repeat the previous 2 steps until you reach the last term.

snthetic division - step 6

  • Separate the last term thus obtained which is the remainder.
  • Now group the coefficients with the variables to get the quotient.

Therefore, the result obtained after synthetic division of (2x 3 - 3x 2 + 4x + 5)/(x + 2) is 2x 2 - 7x + 18 and remainder is -31

How to do Synthetic Division?

Synthetic division of polynomials uses numbers for calculation and avoids the usage of variables . In the place of division, we multiply, and in the place of subtraction, we add.

  • Write the coefficients of the dividend and use the zero of the linear factor in the divisor's place.
  • Bring the first coefficient down and multiply it with the divisor.
  • Write the product below the 2nd coefficient and add the column.
  • Repeat until the last coefficient. The last number is taken as the remainder.
  • Take the coefficients and write the quotient.
  • Note that the resultant polynomial is of one order less than the dividend polynomial.

1) Consider this division: (x 3 - 2x 3 - 8x - 35)/(x - 5). The polynomial is of order 3. The divisor is a linear factor. Let's use synthetic division to find the quotient. Thus, the quotient is one order less than the given polynomial. It is x 2 + 3x + 7 and the remainder is 0. (x 3 - 2x 3 - 8x - 35)/(x - 5) = x 2 + 3x + 7.

Synthetic division of polynomials example

Tips and Tricks on Synthetic Division:

  • Write down the coefficients and divide them using the zero of the linear factor to obtain the quotient and the remainder. (P(x)/(x - a) = Q(x) + (R/(x - a))
  • When we do synthetic division by (bx + a), we should get (Q(x)/b) as the quotient.
  • Perform synthetic division only when the divisor is a linear factor.
  • Perform multiplication and addition in the place of division and subtraction that is used in the long division method.

☛ Related Articles:

  • Long division of polynomials
  • Division Algorithm for Polynomials
  • Dividing Two Polynomials
  • Division of Polynomial by Linear Factor

Synthetic Division Examples

Example 1: The distance covered by Steve in his car is given by the expression 9a 2 - 39a - 30. The time taken by him to cover this distance is given by the expression (a - 5). Find the speed of the car.

Speed is given as the ratio of the distance to the time.

Speed = (9a 2 - 39a - 30)/(a - 5)

synthetic division of polynomial

Speed = (9a + 6)

Answer: Speed is given by the expression 9a + 6.

Example 2: The volume of Sara's storage box is 8x 3 + 12x 2 - 2x - 3. She knows that the area of the box is 4x 2 - 1. What could be the height of the box?

Area (A) = length(l) × breadth(b)

Given A = 4x 2 - 1. This is of the form a 2 - b 2 = (a + b)(a - b)

This can be expressed as, A = (2x + 1)(2x - 1)

V = l × b × h = A × h

h = (V/A) = (8x 3 + 12x 2 - 2x - 3)/[(2x + 1)(2x - 1)]

Let's solve this by the synthetic division twice.

synthetic division of polynomials

Answer: Height of the box = 2x + 3.

Example 3: Perform synthetic division to solve the following expression: (6x 2 + 7x - 20)/(2x + 5).

Let us have a look at the steps shown below,

synthetic division of polynomials example

Answer: Quotient for the given division of polynomials = 3x - 4.

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how to solve a synthetic division problem

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Practice Questions on Synthetic Division

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FAQs on Synthetic Division

When a polynomial has to be divided by a linear factor, the synthetic division is the shortest method. It is an alternative to the traditional long division method used to solve the polynomial division.

How do you Divide Polynomials by Synthetic Division?

We can perform synthetic division using some general steps. Take the coefficients alone, bring the first down, multiply with the zero of the linear factor, and add with the next coefficient and repeat until the end.

What is the Importance of the Synthetic Division?

Synthetic division can be generalized and expanded to the division of any polynomial with any polynomial. It is an easier method in comparison to the long division method for performing division on polynomials with the linear divisor.

What are the Advantages of the Synthetic Division of Polynomials?

This method uses fewer calculations and is quicker than long division. It takes comparatively lesser space while computing the steps involved in the polynomial division.

What are the Disadvantages of Synthetic Division?

The synthetic division can be used only when the divisor is a linear polynomial. We have to follow the long division method for the other cases.

What are the Main Uses of Synthetic Division of Polynomials?

Synthetic division of polynomials helps in finding the zeros of the polynomial. It also reduces the complexity of the expression while dividing the polynomials by a linear factor.

What is the Quotient in Synthetic Division?

In synthetic division, the polynomial obtained is one power lesser than the power of the dividend polynomial. The result obtained can be arranged to form the quotient of the polynomial division.

how to solve a synthetic division problem

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Synthetic Division: The Process

The Process Worked Examples Finding Zeroes Factoring Polynomials

What is synthetic division?

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor — and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.

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How are polynomial zeroes and factors related?

If you are given, say, the polynomial equation y =  x 2  + 5x + 6 , you can factor the polynomial as y = ( x  + 3)( x  + 2) . Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that the two zeroes of the polynomial are x  = −2 and x  = −3 .

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You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that x  = −2 and x  = −3 are the zeroes of a quadratic, then you know that x  + 2 = 0 , so x  + 2 is a factor, and x  + 3 = 0 , so x  + 3 is a factor. Therefore, you know that the quadratic must be of the form y = a ( x  + 3)( x  + 2) .

(The extra number " a " in that last sentence is in there because, when you are working backwards from the zeroes, you don't know toward which quadratic you're working. For any non-zero value of " a ", your quadratic will still have the same zeroes. But the issue of the value of " a " is just a technical consideration; as long as you see the relationship between the zeroes and the factors, that's all you really need to know for this lesson.)

Anyway, the above is a long-winded way of saying that, if x  −  n is a factor, then x  =  n is a zero, and if x  =  n is a zero, then x  −  n is a factor. And this is the fact you use when you do synthetic division.

Let's look again at the quadratic from above: y =  x 2  + 5 x  + 6 . From the Rational Roots Test , we know that ± 1, 2, 3, and 6 are possible zeroes of the quadratic. (And, from the factoring above, we know that the zeroes are, in fact, −3 and −2 .) How would you use synthetic division to check the potential zeroes?

Well, think about how long polynomial divison works. If I were to guess that x  = 1 is a zero, then this means that x  − 1 is a factor of the quadratic. And if it's a factor, then it will divide out evenly; that is, if we divide x 2  + 5 x  + 6 by x  − 1 , we would get a zero remainder. Let's check:

As expected (since we know that x  − 1 is not a factor), we got a non-zero remainder. What does this look like in synthetic division?

How do you do synthetic division?

First, take the polynomial (in our case, x 2  + 5 x  + 6 ), and write the coefficients ONLY inside an upside-down division-type symbol:

Make sure you leave room inside, underneath the row of coefficients, to write another row of numbers later.

Put the test zero, in our case x  = 1 , at the left, next to the (top) row of numbers:

Take the first number that's on the inside, the number that represents the polynomial's leading coefficient, and carry it down, unchanged, to below the division symbol:

Multiply this carry-down value by the test zero on the left, and carry the result up into the next column inside:

Add down the column:

Multiply the previous carry-down value by the test zero, and carry the new result up into the last column:

This last carry-down value is the remainder.

Comparing, you can see that we got the same result from the synthetic division, the same quotient (namely, 1 x  + 6 ) and the same remainder at the end (namely, 12 ), as when we did the long division:

The results are formatted differently, but you should recognize that each format provided us with the same result, being a quotient of x  + 6 , and a remainder of 12 .

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We already know (from the factoring above) that x  + 3 is a factor of the polynomial, and therefore that x  = −3 is a zero.

Now let's compare the results of long division and synthetic division when we use the factor x  + 3 (for the long division) and the zero x  = −3 (for the synthetic division):

As you can see above, while the results are formatted differently, the results are otherwise the same:

In the long division, I divided by the factor x  + 3 , and arrived at the result of x  + 2 with a remainder of zero. This means that x  + 3 is a factor, and that x  + 2 is left after factoring out the x  + 3 . Setting the factors equal to zero, I get that x  = −3 and x  = −2 are the zeroes of the quadratic.

In the synthetic division, I divided by x  = −3 , and arrived at the same result of x  + 2 with a remainder of zero. Because the remainder is zero, this means that x  + 3 is a factor and x  = −3 is a zero. Also, because of the zero remainder, x  + 2 is the remaining factor after division. Setting this equal to zero, I get that x  = −2 is the other zero of the quadratic.

I will return to this relationship between factors and zeroes throughout what follows; the two topics are inextricably intertwined.

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Synthetic Division

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Synthetic division is a shorthand method to find the quotient and remainder when dividing a polynomial by a monic linear binomial \((\)a polynomial of the form \(x-k).\)

\[\frac{x^3-3x^2+5x+6}{x+2} = x^2-5x+15 -\frac{24}{x+2} \\ \]

This process is equivalent to polynomial division , but it requires much less writing. In addition to this application, synthetic division can be used to evaluate a polynomial function at a certain value.

Performing Synthetic Division

Evaluating polynomial functions with synthetic division.

Synthetic division can be used whenever you are dividing a polynomial by a monic linear binomial. A "monic linear binomial" is simply a polynomial of the form \(x-k.\) Examples of monic linear binomials are \(x+2,\) \(x-4,\) and \(x+\frac{4}{3}.\)

Perform \( (x^2 - 3x + 2) \div (x + 1) .\) The top three numbers are the coefficients of the polynomial, that is, 1, -3, and 2 in order. Our "side term" written to the left is based dividing by \( x - k,\) so we write negative rather than positive 1. Most of the last line is the coefficients of the quotient (note the quotient has a degree 1 less than the original polynomial), except the last number which is the remainder. Therefore, \( (x^2 - 3x + 2) \div (x + 1) = x - 4 + \frac{6}{x+1} .\)

Here's the general procedure; we'll follow this with another example.

(Step 1) Write the coefficients of the polynomial as written in standard form, in order. With the example \( x^2 + 2x + 6 ,\) the coefficients are 1, 2, and 6. If there is a "missing term," then one or more 0s must be used. For example, \( 5x^4 + 2x^2 - 5 \) has the coefficients 5, 0, 2, 0, and -5. \(\big(\)Formally, a "missing term" means the polynomial can be written with at least one term \( 0x^k, \) where \(k\) is a non-negative integer less than the degree of the polynomial.\(\big)\)

(Step 2) Draw a vertical line and bar, as shown below.

  • (Step 3) Given that we are dividing by \( x-k ,\) to the left, write \(k.\) \((\)Notice the subtraction; it means if we are dividing by \(x+1,\) i.e. \(k = -1.)\)
  • (Step 4) We start on the far left of the polynomial coefficients and "add" the first coefficient to the number below it—since there's no number, we're "adding" 0, and then write the same number below the line.
  • (Step 5) We take our result and multiply it by the value \(k\) to the left. We then right result diagonally up and to the right from the last position.
  • (Step 6) We continue like step 4 and add the next column (this time there are two numbers to add).
  • (Step 7) We continue like step 5, taking the result of our sum, multiplying by \(k,\) and writing the result diagonally up and to the right.
  • (Step 8) The steps 6 and 7 continue until the last column is reached. The final number written will be our remainder. \((\)If the remainder is 0, that means \(x-k\) was a factor of our original polynomial.\()\) The other numbers on the bottom row represent the result of multiplication; they are the coefficients of the quotient's polynomial, and the degree of the quotient's polynomial is 1 less than the degree of the original polynomial.
Do the division \((3x^4 - 2x^2 + 3) \div (x - 2).\) Note that \( 3x^4 - 2x^2 + 3 = 3x^4 + 0x^3 - 2x^2 + 0x + 3 ,\) and we need the "missing terms" when writing out the synthetic division. We are dividing by \( x - k = x - 2, \) so \( k = 2:\) This means \( \frac{3x^4 - 2x^2 + 3}{x-2} = 3x^3 + 6x^2 + 10x + 20 + \frac{43}{x-2} .\) \(_\square\)

If you are curious why the algorithm works, compare the regular long division algorithm with the synthetic. Below the numbers that serve the same function are marked. The synthetic division diagram simply collapses the operations that would normally happen when dividing by \( (x-k) \) into a format simpler to write.

Note that we'd often do long division by multiplying and then subtracting, but because we've already reversed the sign of our divisor \(\big(\)by using \( (x-k) \) instead of \( (x+k)\big),\) we've accounted for this; in other words, multiplying the second term by -1 and adding is the same as subtracting. \(\big(\)In fact, it'd be possible to change the synthetic division algorithm to use subtraction and have the side term be \( (x+k),\) but the examples above follow the traditional form of the algorithm.\(\big)\)

Find the remainder when \(x^{3} + 4x^{2} - 5x + 3 \) is divided by \(x-2\).

Main Article: Remainder Factor Theorem

By applying the remainder factor theorem , we can evaluate polynomial functions with synthetic division.

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Synthetic Division

In these lessons, we will look at Synthetic Division, which is simplified form of long division.

Related Pages Long Division Of Polynomials More Lessons for Algebra Math Worksheets

What is Synthetic Division?

Synthetic Division is an abbreviated way of dividing a polynomial by a binomial of the form ( x + c ) or ( x – c ). We can simplify the division by detaching the coefficients.

Example: Evaluate ( x 3 – 8 x + 3) ÷ ( x + 3) using synthetic division

Solution: ( x 3 – 8 x + 3) is called the dividend and ( x + 3) is called the divisor.

Step 1: Write down the constant of the divisor with the sign changed –3

Step 2: Write down the coefficients of the dividend. (Remember to add a coefficient of 0 for the missing terms)

how to solve a synthetic division problem

Step 3: Bring down the first coefficient.

how to solve a synthetic division problem

Step 4: Multiply (1)( –3) = –3 and add to the next coefficient.

how to solve a synthetic division problem

Repeat Step 4 for all the coefficients

synthetic division example

We find that ( x 3 – 8 x + 3) ÷ ( x + 3) = x 2 – 3 x + 1

It is easier to learn Synthetic Division visually. Please watch the following videos for more examples of Synthetic Division.

Polynomial Division: Synthetic Division Perform synthetic division to divide by a binomial in the form (x - k)

Example: Divide using synthetic division

(2x 3 + 6x 2 + 29) ÷ (x + 3)

(2x 3 + 6x 2 - 17x + 15) ÷ (x + 5)

(y 5 - 32) ÷ (y - 2)

(16x 3 - 2 + 14x - 12x 2 ) ÷ (2x + 1)

Divide a Trinomial by a Binomial Using Synthetic Division

(x 2 - 5x + 7) ÷ (x - 2)

(x 2 + 8x + 12) ÷ (x + 2)

Synthetic Division This video shows how you can use synthetic division to divide a polynomial by a linear expression. It also shows how synthetic division can be used to evaluate polynomials.

Example: (x 3 - 2x 2 + 3x - 4) ÷ (x - 2)

Synthetic Division This video shows how to use synthetic division to divide a polynomial by a linear expression and also how to use the remainder to evaluate the polynomial.

Example: (x 4 - x 2 + 5) ÷ (x + 3)

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Chapter 3: Polynomial Functions

3.4.2: synthetic division, learning outcomes.

  • Use synthetic division to divide polynomials by a linear binomial.

Although long division of polynomials will always work, there is a shorthand method for the special case of dividing a polynomial by a linear factor whose leading coefficient is 1. This shorthand method is called synthetic division .

Consider the example of dividing [latex]2{x}^{3}-3{x}^{2}+4x+5[/latex] by [latex]x+2[/latex] using the long division algorithm. Work through the example on your own to make sure you know how to get the solution in figure 1.

Division of a polynomial by a linear binomial

Figure 1. Division of a polynomial by a linear binomial

There is a lot of repetition when we use the division algorithm. If we don’t write the variables but instead line up their coefficients in columns under the division sign and also eliminate some partial products, we already have a simpler version of the entire problem.

Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.

Figure 2. Losing the variables

Synthetic division takes this simplification further. Collapse the algorithm by moving each of the rows up to fill any vacant spots. Also, instead of multiplying and subtracting the product, we change the sign of the “divisor” to –2, so we can add rather than subtract. The process starts by bringing down the leading coefficient.  We then multiply it by the “divisor” and add, repeating this process column by column until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[/latex] and the remainder is –31. 

Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.

Figure 3. Synthetic division

synthetic division

Synthetic division is a shortcut that can be used when the divisor is a binomial in the form [latex]x–k[/latex], where [latex]k[/latex] is a constant. In  synthetic division , only the coefficients are used in the division process.

  • Write [latex]k[/latex]   for the divisor.
  • Write the coefficients of the dividend written in descending order.
  • Bring the leading coefficient down in the first column.
  • Multiply the leading coefficient by [latex]k[/latex]. Write the product in the next column.
  • Add the terms of the second column.
  • Multiply the result by [latex]k[/latex]. Write the product in the next column.
  • Repeat steps 5 and 6 for the remaining columns.
  • Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on. The degree of the quotient will be one degree less than the degree of the dividend.

There are then two basic rules to synthetic division:

  • Add vertically
  • Multiply diagonally by [latex]k[/latex]

The process will be made clearer in the examples that follow.

Use synthetic division to divide [latex]5x^2-3x-36[/latex] by [latex]x - 3[/latex].

Begin by setting up the synthetic division. Determine the value of [latex]k[/latex]: [latex]x-3=x-k[/latex], so [latex]k=3[/latex].

Write [latex]k=3[/latex] and the coefficients of the dividend in descending order.

Bring down the leading coefficient. Multiply the leading coefficient 5 by [latex]k=3[/latex] and put the answer under the second column.

The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.

Add the numbers vertically in the second column.

Multiply the resulting number by [latex]k=3[/latex] and write the result in the next column: [latex]12\cdot 3=36[/latex].

Then add the numbers vertically in the third column.

Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.

The quotient is [latex]5x+12[/latex] with remainder is 0.

[latex]\dfrac{5x^2-3x-36}{x-3}=5x+12[/latex]

With no remainder, [latex]x - 3[/latex] and [latex]5x+12[/latex] are factors of the original polynomial.

[latex]5x^2-3x-36=(x-3)(5x+12)[/latex]

Study Guides > College Algebra CoRequisite Course

Synthetic division, learning outcomes.

  • Use synthetic division to divide polynomials.

cnx_precalc_revised_eq_42

A General Note: Synthetic Division

How to: given two polynomials, use synthetic division to divide.

  • Write k  for the divisor.
  • Write the coefficients of the dividend.
  • Bring the leading coefficient down.
  • Multiply the leading coefficient by k . Write the product in the next column.
  • Add the terms of the second column.
  • Multiply the result by k . Write the product in the next column.
  • Repeat steps 5 and 6 for the remaining columns.
  • Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.

Example: Using Synthetic Division to Divide a Second-Degree Polynomial

A collapsed version of the previous synthetic division.

Analysis of the Solution

Example: using synthetic division to divide a third-degree polynomial.

Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.

Example: Using Synthetic Division to Divide a Fourth-Degree Polynomial

.

Answer: [latex]3{x}^{3}-3{x}^{2}+21x - 150+\frac{1,090}{x+7}[/latex]

Example: Using Polynomial Division in an Application Problem

Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.

[latex]\begin{array}{l}V=l\cdot w\cdot h\\ 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\cdot \left(x - 2\right)\cdot h\end{array}[/latex]

[latex]\begin{array}{l}\frac{3x\cdot \left(x - 2\right)\cdot h}{3x}=\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x}\\ \left(x - 2\right)h={x}^{3}-{x}^{2}-11x+18\end{array}[/latex]

[latex]h=\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[/latex]

Synthetic division with 2 as the divisor and {1, -1, -11, 18} as the quotient. The result is {1, 1, -9, 0}

Answer: [latex]3{x}^{2}-4x+1[/latex]

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Understanding the Remainder in a Synthetic Division Problem

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When studying higher level mathematics, such as calculus and algebra, synthetic division can help simplify quite complex divisions and processes. This is a valuable tool to help speed up the process and get to the desired answer more quickly. However, when performing a synthetic division problem, you must be able to correctly calculate the remainder for the problem for it to be accurate. In this article, we will discuss what synthetic division is, how to perform a synthetic division problem, working through an example of a synthetic division problem, finding the remainder in a synthetic division problem, tips for solving synthetic division problems and common mistakes to avoid when performing synthetic division.

What is Synthetic Division?

Synthetic division is a method of simplifying complex polynomial long division problems. It is a simplified method of the traditional long division process and can often give you the desired result more quickly and easily than using the standard long division process. It is useful for solving polynomials with a maximum degree (the largest exponent sum of all its terms) of 3 or 4. Occasionally, it can also solve polynomials with a degree of 5 if the highest degree terms are simple.

How to Perform a Synthetic Division Problem

Synthetic division follows a set structure to ensure accuracy. It requires the following steps:

  • Write down the polynomial you want to divide in typical polynomial form, with the highest degree (or “leading”) term first and each lower degree term following consecutively.
  • Write the divisor (either an integer or polynomial) to the right of the polynomial, separated by two vertical lines.
  • Write 0’s in every row below the coefficients from left to right.
  • Divide each coefficient of the dividend by the divisor. Then for each row below it, multiply it by the divisor and add it to the next coefficient.
  • Continue this process until all rows in this table are filled out. The number at the bottom is your remainder.

Working Through an Example Synthetic Division Problem

Now let’s look at an example problem to help demonstrate how this works. Say you want to divide x³-3x²+3x+7 by x-2 . Here are the steps you should take:

  • Write x³-3x²+3x+7 by its highest degree to lowest degree terms: x³-3x²+3x+7 . Also write the divisor x-2 separated by two vertical lines.
  • Fill in the empty rows below each coefficient with 0’s, starting from left to right.
  • Divide the first coefficient (1) by the first coefficient of the divisor (1). This equals 1.
  • Multiply the 1 in the divisor column by -3 and add it to the -3 in the second row below it (1 x -3 = -3 -3 = -6). Then divide this by the first coefficient of the divisor (1). This equals -6.
  • Multiply the 1 in the divisor column by 3 and add it to the 3 in the third row below it (-6 x 1 = -6 + 3 = -3). Divide this by 1. This equals -3.
  • Multiply the 1 in the divisor column by 7 and add it to 0 in in fourth row below it (-3 x 1 = -3 + 7 = 4). Divide this by 1. This equals 4.

The result of this synthetic division problem would be (x²-6x-3) + 4 . The nice thing about synthetic division is that you don’t have to spend much time writing down long division steps as you might have with traditional long division—it’s all compressed into one step. The 4 is your remainder for this problem and can either be left in fraction form or converted into a decimal if desired.

Finding the Remainder in a Synthetic Division Problem

Now that we have an idea of how a synthetic division problem works, let’s look at how you can find the remainder specifically. Essentially, Synthetic Division works by expanding out each of the coefficients and numbers that make up the dividend, then breaking up each coefficient into polynomials spanning certain degrees until you reach your remainer. To find your remainder in a Synthetic Division problem:

  • Write down your dividend and divisor with two vertical lines between them.
  • Start with the last number in your dividend and divide by the first number in your divisor. Whatever your result is, that number is your remainder.

Tips for Solving Synthetic Division Problems

Synthetic division can help speed up complex polynomial long division processes, but it’s important to understand all of its steps before jumping into a problem. Here are some helpful tips for solving synthetic division problems:

  • Start with simple practice examples before jumping straight into complex problems.
  • Always double check your work to make sure no mistakes were made while breaking up coefficients or multiplying numbers.
  • Draw out visual diagrams if needed to help break up coefficients into polynomials more easily.
  • When solving more complex problems, use a calculator if available for extra accuracy.

Common Mistakes to Avoid When Performing Synthetic Division

It’s important to be mindful of potential mistakes that may occur when performing synthetic divisions. Here are some common mistakes to avoid when performing synthetic division:

  • Not breaking down coefficients into more specific degrees before performing synthetic division.
  • Failing to accurately multiply numbers while filling out rows.
  • Not verifying your answer after completing synthesis division—always double check your work!

Resources for Further Learning About Synthetic Division

There are many online resources available to help you further understand how synthetic division works and practice performing these type of problems. Here are some great resources you can use:

  • Math Is Fun: Synthetic Division
  • Khan Academy: Synthetic Division
  • Purple Math: Synthetic Division Calculator
  • Algebra Online: Synthetic Division Rules & Examples

By understanding how Synthetic Division works and avoiding common mistakes, you can easily figure out how to find remainders in these problems and solve them accurately and quickly in no time!

how to solve a synthetic division problem

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COMMENTS

  1. Synthetic Division (Definition, Steps and Examples)

    The process of the synthetic division will get messed up if the divisor of the leading coefficient is other than one. In case if the leading coefficient of the divisor is other than 1 while performing the synthetic division method, solve the problem carefully. The basic Mantra to perform the synthetic division process is"

  2. Synthetic Division

    Synthetic Division Method. I must say that synthetic division is the most "fun" way of dividing polynomials. It has fewer steps to arrive at the answer as compared to the polynomial long division method.In this lesson, I will go over five (5) examples that should hopefully make you familiar with the basic procedures in successfully dividing polynomials using synthetic division.

  3. Intro to polynomial synthetic division (video)

    Synthetic division reminds me of a FOIL - a thing from algebra to remember on how to distribute something like: (x+4)(x-3). Problem with Synthetic division and FOIL, is that they work only in couple simple cases and not in complex situations. Edit: Apparently, I was wrong to some extent.

  4. Synthetic Division

    The following are the steps while performing synthetic division and finding the quotient and the remainder. We will take the following expression as a reference to understand it better: (2x 3 - 3x 2 + 4x + 5)/(x + 2). Check whether the polynomial is in the standard form.; Write the coefficients in the dividend's place and write the zero of the linear factor in the divisor's place.

  5. 2.7: Synthetic Division

    Synthetic Division is a handy shortcut for polynomial long division problems in which we are dividing by a linear polynomial. This means that the highest power of \(x\) we are dividing by needs to be \(x^{1}\). This limits the usefulness of Synthetic Division, but it will serve us well for certain purposes.

  6. Dividing polynomials: synthetic division (video)

    Well you could technically use 3x-3 for synthetic division because if you set that expression equal to zero, then you get 3x-3=0. Then add 3 to both sides, 3x=3. After that divide both sides by 3 to get the coefficient off the x term, x=1. But for denominator expressions where you can't do what I just did, you would need to use long division.

  7. Synthetic Division

    How To: Given two polynomials, use synthetic division to divide. Write k for the divisor. Write the coefficients of the dividend. Bring the leading coefficient down. Multiply the leading coefficient by k . Write the product in the next column. Add the terms of the second column. Multiply the result by k . Write the product in the next column.

  8. How does synthetic division of polynomials work?

    Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor — and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.

  9. Synthetic Division

    Synthetic division is a shorthand method to find the quotient and remainder when dividing a polynomial by a monic linear binomial \((\)a polynomial of the form \(x-k).\) \[\frac{x^3-3x^2+5x+6}{x+2} = x^2-5x+15 -\frac{24}{x+2} \\ \] This process is equivalent to polynomial division, but it requires much less writing.In addition to this application, synthetic division can be used to evaluate a ...

  10. Synthetic Division (examples, solutions, videos)

    We can simplify the division by detaching the coefficients. Example: Evaluate ( x3 - 8 x + 3) ÷ ( x + 3) using synthetic division. Solution: ( x3 - 8 x + 3) is called the dividend and ( x + 3) is called the divisor. Step 1: Write down the constant of the divisor with the sign changed. -3.

  11. 3.4.2: Synthetic Division

    Solution. Begin by setting up the synthetic division. Determine the value of k k: x−3 =x−k x − 3 = x − k, so k= 3 k = 3. Write k =3 k = 3 and the coefficients of the dividend in descending order. Bring down the leading coefficient. Multiply the leading coefficient 5 by k= 3 k = 3 and put the answer under the second column.

  12. Process of SYNTHETIC DIVISION

    This video tutorial provides a basic introduction to the process of synthetic division of polynomials. You can use it to find the quotient and remainder of ...

  13. Synthetic Division

    We'll go step by step and solve the same problem we did earlier, this time using synthetic division. Step 1: Make sure the terms of the numerator are in descending order. If a term is missing, add it in with a coefficient of 0. Step 2: Set the denominator equal to 0 and solve to find the number to put as the divisor.

  14. Synthetic Division

    This video shows how to set-up and solve synthetic division of a polynomial. It starts with a quick problem using long division and compares it to its synthe...

  15. Synthetic Division of Polynomials

    This precalculus video tutorial provides a basic introduction into synthetic division of polynomials. You can use it to find the quotient and remainder of a...

  16. Study Guide

    Use synthetic division to divide 5 {x}^ {2}-3x - 36 5x2 −3x−36 by x - 3 x−3 . Answer: Begin by setting up the synthetic division. Write k and the coefficients. Bring down the leading coefficient. Multiply the leading coefficient by k . Continue by adding the numbers in the second column. Multiply the resulting number by k .

  17. College Algebra Tutorial 37

    Step 1: Set up the synthetic division. Long division would look like this: Synthetic division would look like this: Step 2: Bring down the leading coefficient to the bottom row. *Bring down the 2. Step 3: Multiply c by the value just written on the bottom row. * (-1) (2) = -2 *Place -2 in next column.

  18. Using Synthetic Division to Find the Quotient

    To better understand how synthetic division works, let's look at a few examples. The first example is the problem 2x^4-5x^3+6x^2-7x+10 divided by 2x+1. This problem can be solved by first writing out the divisor, followed by writing out the dividend, zeroes, and another copy of the divisor. The next step is to multiply each term in the ...

  19. Using Synthetic Division to Solve an Equation

    Watch more videos on http://www.brightstorm.com/math/algebra-2SUBSCRIBE FOR All OUR VIDEOS!https://www.youtube.com/subscription_center?add_user=brightstorm2V...

  20. Using Synthetic Division Practice

    Practice Using Synthetic Division with practice problems and explanations. Get instant feedback, extra help and step-by-step explanations. Boost your Algebra grade with Using Synthetic Division ...

  21. Limit with synthetic division

    This problem shown in this video requires factoring a cubic polynomial. The instructor shows how to factor the polynomial using synthetic division and how t...

  22. Understanding the Remainder in a Synthetic Division Problem

    It is useful for solving polynomials with a maximum degree (the largest exponent sum of all its terms) of 3 or 4. Occasionally, it can also solve polynomials with a degree of 5 if the highest degree terms are simple. How to Perform a Synthetic Division Problem. Synthetic division follows a set structure to ensure accuracy.

  23. How to Solve Synthetic Division & Remainder Theorem Word Problems 3

    In this video, I show you how to use polynomial synthetic division to solve word problems. Purchase lesson worksheets with full answer solutions here: https:...