1. Synthetic Evaluation of Polynomials Be able to use synthetic division to evaluate polynomials.

2. FHSPolynomials2 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.

3. FHSPolynomials3 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.

4. FHSPolynomials4 Example 1 Evaluate for x = 4 1. 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.

5. FHSPolynomials5 Example 1 Evaluate for x = 4 1. 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. 4 1 -3 5 -5 1 4 1 4 9 36 31

6. FHSPolynomials6 Example 2 Evaluate for x = – 1 1. 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. 4 -5 0 3 4 -4 -9 9 9 -6