Synthetic Evaluation of Polynomials

Be able to use synthetic division to evaluate polynomials.

FHS Polynomials 2

Evaluating a Polynomial

• To evaluate a polynomial for a given value of x, you can just substitute the value into the polynomial for the x.

• Then you simplify the polynomial to find the answer.

• Sometimes this is rather complicated, so there is a shorter way to evaluate a polynomial.

FHS Polynomials 3

Synthetic Substitution

• Using the remainder theorem gives you an easy way to evaluate a polynomial for a given value (a).

• It states that if you use synthetic division to divide a polynomial by that given value (a), the remainder is the answer to the evaluation of the polynomial for that given value a.

FHS Polynomials 4

Example 1

Evaluate for x = 41. To use substitution to evaluate the

polynomial for the value of 4, plug 4 into the polynomial for x.

2.

3. This is one way to work the problems, but sometimes it is easier to use the remainder theorem and use synthetic division.

3 23 5 5 x x x

3 24 3 4 5 4 5

64 48 20 5 31

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Example 1

Evaluate for x = 41. Write 4 in the upper right corner and then

write the coefficients of the polynomial. 2. Continue to divide using synthetic

division.

3. Since the remainder is 31, the answer to the problem is 31.

3 23 5 5 x x x

4 1 -3 5 -5

1

4

1

4

9

36

31

FHS Polynomials 6

Example 2

Evaluate for x = – 11. Write –1 in the upper right corner and

then write the coefficients of the polynomial.

2. Continue to divide using synthetic division.

3. Since the remainder is -6, the answer to the problem is -6.

3 24 5 3x x

-1 4 -5 0 3

4

-4

-9

9

9

-9

-6