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Synthetic Division – Explanation & Examples

A polynomial is an algebraic expression made up of two or more terms which are subtracted, added or multiplied. A polynomial can contain coefficients, variables, exponents, constants and operators such addition and subtraction.

It is also important to note that, a polynomial can’t have fractional or negative exponents. Examples of polynomials are; 3y2 + 2x + 5, x3 + 2 x 2 − 9 x – 4, 10 x 3 + 5 x + y, 4x2 – 5x + 7) etc. Like number, polynomials can undergo addition, subtraction, multiplication, and division.

We saw addition, subtraction, multiplication, and long division of polynomials previously, let’s have a look at synthetic division now.

There are two methods in mathematics for dividing polynomials.

These are the long division and the synthetic method. As the name suggests, the long division method is most cumbersome and intimidating process to master. On the other hand, the synthetic method is a “fun” way of dividing polynomials.

I must say that synthetic division is a shortcut way to divide polynomials because it entails  fewer steps to arrive at the answer as compared to polynomial long division method. In this article, we are going to learn what synthetic division method is and how to do method with a couple of examples.

What is Synthetic Division?

Synthetic division can be defined as a shorthand way of dividing one polynomial by another polynomial of first degree. The synthetic method involves finding zeroes of the polynomials.

How to do Synthetic Division?

To divide a polynomial using synthetic division, you should divide it with a linear expression whose leading coefficient must be 1.

This type of division by a linear denominator is commonly known as division by Ruffini’s rule or the “paper-and-pencil computation.”

For the synthetic division method to be possible, the following requirements must be meet:

  • The divisor should be a linear factor. This means that, the divisor should be an expression of degree 1.
  • The leading coefficient of the divisor should also be 1. If the coefficient of the divisor is other than 1, the synthetic division process will get messed up. Therefore, you will be forced to manipulate the divisor in order to convert the leading coefficient to 1. For example, 4x – 1 and 4x + 9 would be x – ¼ and x + 9/4 respectively.

To perform polynomial synthetic division, here are the steps:

  • Set the divisor to zero in order to find the number to put in the division box.
  • Express the dividend in standard form. This is the same as writing the dividend in descending order. If the dividend is missing some terms, fill them using zero. For example, 3x4 + 2 x3 + 3x2 + 5 = 3x4 + 2 x3 + 3x2 + 0x +5
  • Now, bring down the leading coefficient in the dividend.
  • Place in the preceding column the product of the number you brought down and the number in the division box.
  • Write result in the bottom of the row by adding the product from step 4 and preceding number.
  • Repeat procedure 5 until the remainder is zero or a numerical value.
  • Write your final answer as the numbers in the bottom column. When there is a remainder in division box, express it as a fraction with its denominator being the divisor.

NOTE: The variable in the answer is one power less than the original dividend

You can master the above steps by using the following mantra: “Bring down, Multiply and add, multiply and add, Multiply and add, ….”

Example 1

Divide x3 + 5x2 -2x – 24 by x – 2

Solution

Change the sign of constant in the divisor x -2 from -2 to 2 and drop it down.

_____________________
x – 2     | x ³   + 5x²   –   2x   –  24

2   |    1        5         -2       -24

Also bring down the leading coefficient. This means that 1 be the first number of the quotient.

2 |    1        5        -2       -24
________________________
1

Multiply 2 by 1 and add 5 to the product to get 7. Now bring 7 down.

2 |    1        5        -2       -24
2
________________________
1        7

Multiply 2 by 7 and add – 2 to the product to get 12. Bring 12 down

2 |    1        5        -2       -24
2        14
__________________________
1        7        12

Finally, multiply 2 by 12 and add -24 to the result to get 0.

2 |    1        5        -2       -24
2        14       24
__________________________
1        7        12         0

Hence;

x3 + 5x2 -2x – 24/ x – 2 = x² + 7x + 12

Example 2

Divide x2 + 11x + 30 by x + 5

Solution

Change the sign of constant in the divisor x + 5 from 5 to -5 and bring it down.

_____________________
x + 5    | x2 + 11x + 30

-5 |    1    11      30

Bring down the coefficient of the first term in dividend. This will be our first quotient

2 |    1        11        30
________________________
1

Multiply -5 by 1 and add 11 to the product to get 6. Bring 6 down;

-5 |    1        11        30
-5
________________________
1        6

Multiply -5 by 6 and add 30 to the result to get 0.

-5 |    1        11        30
-5        -30
________________________
1        6            0

Therefore, the quotient is x + 6

Example 3

Divide 2x3 + 5x2 + 9 by x + 3

Solution

Reverse the sign of constant in the divisor x + 3 from 3 to -3 and bring it down.

_____________________
x + 3    | 2x3 + 5x2 + 0x + 9

-3|    2      5       0    9

Bring down the coefficient of the first term in dividend. This will be our first quotient

-3 |    2        5        0     9
________________________
2

Multiply -3 by 2 and add 5 to the product to get -1. Bring -1 down;

-3 |    2       5    0    9
-6
________________________
2        -1

Multiply -3 by -1 and add 0 to the result to get 3. Bring 3 down.

-3 |    2        5         0      9
-6       3
________________________
2        -1          3

Multiply -3 by 3 and add -9 to the result to get 0.

-3 |    2        5         0      9
-6       3      -9
________________________
2        -1          3     0

Therefore, 2x2– x + 3 is the correct answer.

Example 4

Use synthetic division to divide ​3​​x3 + 10x​2 ​​− 6x −20 by x+2.

Solution

Reverse the sign of x + 2 from 2 to -2 and bring it down.

_____________________
x + 2   |4x3 + 10x​2 ​​− 6x − 20

-2| 4     10       6      20

Bring down the coefficient of the first term in dividend.

-2 |    4        10        6     20
________________________
4

Multiply -2 by 4 and add 10 to get 2. Bring 2 down;

-2 |    4        10        6     20
-8
________________________
4        2

Multiply -2 by 2 and add -6 to the result to get 10. Bring -10 down.

-2 |   4        10        -6     20
-8       -4
________________________
4       2         10

Multiply -2 by 10 and add 20 to the result to get 0.

-2 |   4        10        -6     20
-8       -4     -20
________________________
4       2         -10        0

Therefore, 4x​2 ​​+ 2x −10 is the answer.

Example 5

Divide ​-9x4 +10x​3 ​​+ 7x​2 ​​− 6 by x−1.

Solution

​-9x4 +10x​3 ​​+ 7x​2 ​​− 6 / x−1 =

1 |   -9        10        7       0    -6
-8        1     8      8
________________________
-9       8        8     2

Therefore, the answer is -9x3 +8x2+8x + 2/x -1

Practice Questions

Use synthetic division to divide the following polynomials:

  1. 2x3 – 5x2 + 3x + 7 by x -2
  2. x3 – 5x2 + 3x +7 by x -3
  3. 2x3 + 5x2 + 9 by x + 3
  4. x5 – 3x3 – 4x – 1 by x -1
  5. – 2x4 + x by x -3
  6. – x5 + 1 by x + 1
  7. 2x3 – 13x2 + 17x – 10 by x – 5
  8. x4 – 3x3 – 11x2 + 5x + 17 by x + 2
  9. 4x3 – 8x2 – x + 5 by 2x -1

Answers

  1. 2x2 – x + 1 + 9/x-2
  2. x2 – 2x -2 -2/x-3
  3. 2x2 – x + 3 + 3/x + 3
  4. x4 + x3 – 2x2 – 2x – 7/x-1
  5. -2x3 – 6x2 – 18x -53 – 159/x-3
  6. -x4 + x3 – x2 + x – 1 + 2/x + 1
  7. 2x2 – 3x + 2
  8. x3 – 5x2 – x + 7 + 3/x + 2
  9. 4x2 -6x -4 + 3/ (x – ½)

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