Evaluate a polynomial by synthetic division

Learn how to evaluate a polynomial by synthetic division with a couple of well-chosen examples.

Example #1:

Find P(3) using synthetic division for P(x) = x⁴ - 3x³ + 5x² + 6x + 2

By the remainder theorem, P(3) is equal to the remainder when P(x) is divided by x - 3

Step 1

Start by writing down the divisor and the coefficients of the polynomial in standard form.

3 | 1 -3 5 6 2

Step 2

Bring down the first coefficient and that is 1.

3 | 1 -3 5 6 2

________________
1

Step 3

Multiply the first coefficient by the divisor. Write the result under the next coefficient and add.

3 | 1 -3 5 6 2
3
________________
1 0

Step 4

Multiply the 0 by the divisor. Write the result under the next coefficient and add.

3 | 1 -3 5 6 2
3 0
________________
1 0 5

Step 5

Repeat the steps of multiplying and adding until the remainder is found.

3 | 1 -3 5 6 2
3 0 15 63
___________________
1 0 5 21 65

Since the remainder is 65, P(3) = 65

Example #2:

Find P(-2) using synthetic division for P(x) = 2x⁴ + 3x³ - 5x² - 20

By the remainder theorem, P(-2) is equal to the remainder when P(x) is divided by x - (-2)

Step 1

Start by writing down the divisor and the coefficients of the polynomial in standard form.

-2 | 2 3 -5 0 -20

Step 2

Bring down the first coefficient and that is 2.

-2 | 2 3 -5 0 -20

__________________
2

Step 3

Multiply the first coefficient by the divisor. Write the result under the next coefficient and add.

-2 | 2 3 -5 0 -20
-4
___________________
2 -1

Step 4

Multiply the -1 by the divisor. Write the result under the next coefficient and add.

-2 | 2 3 -5 0 -20
-4
___________________
2 -1

Step 5

Repeat the steps of multiplying and adding until the remainder is found.

-2 | 2 3 -5 0 -20
-4 2 6 -12
___________________
2 -1 -3 6 -32

Since the remainder is -32, P(-2) = -32

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