1. 7.4 The Remainder and Factor Theorems
Objectives:
Evaluate functions using synthetic division.
Determine whether a binomial is a factor of a polynomial by using synthetic substitution.

2. Recall synthetic division
Use synthetic division to divide

3. Comparison
Compare the remainder to the answer you get by plugging -1 in for all x’s. (-1)⁴+3(-1)³-8(-1)²+5(-1) =-21 Synthetic division can also be called synthetic substitution because the remainder is the same as plugging in that value in. Which is shorter? Personal preference!

4. Remainder Theorem
If a polynomial f(x) is divided by x – a, the remainder is the constant f(a), and
f(x) = q(x) • (x – a) + f(a) where q(x) is a polynomial with degree one less than the degree of f(x).
When synthetic division is used to evaluate a function, it is called synthetic substitution.

5. Try one Use either method – synthetic substitution or direct substitution.
Evaluate x4 + 5x3 + 5x2 – x + 2 for x =
(-2)⁴+5(-2)³+5(-2)²-(-2)+2= 16+5(-8)+5(4)+2+2= =0 Since the remainder is 0, that means that x+2 is a factor of the original polynomial.

6. Factor Theorem
The binomial x – a is a factor of the polynomial f(x) if and only if f(a) = 0. Example: Is (x-4) a factor of x³-x²-13x+4 Use remainder theorem or substitution. Remember to use the opposite of the potential factor x-a . Use 4 in this case. 4³-(4)²-13(4)+4= =0 Yes, it is a factor.

7. Finding other factors
If one factor is known, the others can be found using synthetic division and then factoring the quotient. Example: Given one factor, find the remaining factors. x³+4x²-15x-18; x factor: (x+6)(x+1) All factors: (x-3)(x+6)(x+1)

8. Try one
x³-9x²+27x-27; x-3

9. Homework p even