# 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.

|  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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