111519 Remainder and Factor Theorems You can use

11/15/19

Remainder and Factor Theorems

You can use synthetic division to evaluate polynomials. The process is exactly the same as the process of synthetic division, but the final answer is interpreted differently, as described by the Remainder Theorem.

The Remainder Theorem Remember Lesson for Synthetic Division And f(4)=33

Example 3 A: Use synthetic substitution to evaluate the polynomial for the given value. P(x) = 2 x 3 + 5 x 2 – x + 7 for x = 2. 2 2 5 – 1 2 4 9 7 18 34 17 41 Write the coefficients of the dividend. Use a = 2. P(2) = 41 Check Substitute 2 for x in P(x) = 2 x 3 + 5 x 2 – x + 7. P(2) = 2(2)3 + 5(2)2 – (2) + 7 P(2) = 41

Example 3 B: Use synthetic substitution to evaluate the polynomial for the given value. 1 P(x) = 6 x 4 – 25 x 3 – 3 x + 5 for x = –. 3 1 Write the coefficients of 6 – 25 0 – 3 5 – 3 the dividend. Use 0 for – 2 9 – 3 2 the coefficient of x 2. Use -1 6 – 27 9 – 6 7 a=. 3 P( -1) = 7 3

Check It Out! Example 3 a Use synthetic substitution to evaluate the polynomial for the given value. P(x) = x 3 + 3 x 2 + 4 for x = – 3 1 3 0 4 – 3 0 0 1 0 0 4 P(– 3) = 4 Write the coefficients of the dividend. Use 0 for the coefficient of x 2 Use a = – 3. Check Substitute – 3 for x in P(x) = x 3 + 3 x 2 + 4. P(– 3) = (– 3)3 + 3(– 3)2 + 4 P(– 3) = 4

Check It Out! Example 3 b Use synthetic substitution to evaluate the polynomial for the given value. P(x) = 5 x 2 + 1 5 9 5 10 P( 1 ) = 5 5 9 x + 3 for x = 3 2 5 1. 5 Write the coefficients of 1 the dividend. Use a =. 5