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