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

2. Example 3A: Using Synthetic Substitution
Use synthetic substitution to evaluate the polynomial for the given value.
P(x) = 2x3 + 5x2 – x + 7 for x = 2.
Check Substitute 2 for x in P(x) = 2x3 + 5x2 – x + 7.
P(2) = 2(2)3 + 5(2)2 – (2) + 7
P(2) = 41 

3. Check It Out! Example 3a
Use synthetic substitution to evaluate the polynomial for the given value.
P(x) = x3 + 3x2 + 4 for x = –3.
Check Substitute –3 for x in P(x) = x3 + 3x2 + 4.
P(–3) = (–3)3 + 3(–3)2 + 4
P(–3) = 4 

4. Objectives
Use the Factor Theorem to determine factors of a polynomial.
Factor the sum and difference of two cubes.

5. Recall that if a number is divided by any of its factors, the remainder is 0. Likewise, if a polynomial is divided by any of its factors, the remainder is 0.
The Remainder Theorem states that if a polynomial is divided by (x – a), the remainder is the value of the function at a. So, if (x – a) is a factor of P(x), then P(a) = 0.

6. Check It Out! Example 1
Determine whether the given binomial is a factor of the polynomial P(x).
a. (x + 2); (4x2 – 2x + 5)
b. (3x – 6); (3x4 – 6x3 + 6x2 + 3x – 30)

7. You are already familiar with methods for factoring quadratic expressions. You can factor polynomials of higher degrees using many of the same methods.

8. Check It Out! Example 2a
Factor: x3 – 2x2 – 9x + 18.

9. Check It Out! Example 2a Continued
Check Use the table feature of your calculator to compare the original expression and the factored form.
The table shows that the original function and the factored form have the same function values. 

10. Check It Out! Example 2b
Factor: 2x3 + x2 + 8x + 4.

11. Just as there is a special rule for factoring the difference of two squares, there are special rules for factoring the sum or difference of two cubes.

12. Check It Out! Example 3a
Factor the expression.
8 + z6

13. Check It Out! Example 3b
Factor the expression.
2x5 – 16x2

14. Check It Out! Example 4
The volume of a rectangular prism is modeled by the function V(x) = x3 – 8x2 + 19x – 12, which is graphed below. Identify the values of x for which V(x) = 0, then use the graph to factor V(x).
V(x) has three real zeros at x = 1, x = 3, and x = 4. If the model is accurate, the box will have no volume if x = 1, x = 3, or x = 4.

15. Check It Out! Example 4 Continued
One corresponding factor is (x – 1). 1
1 –8 19 –12
Use synthetic division to factor the polynomial. 1 –7 12 1 –7 12
V(x)= (x – 1)(x2 – 7x + 12)
Write V(x) as a product.
V(x)= (x – 1)(x – 3)(x – 4)
Factor the quadratic.