1. Divide using long division.
÷ 7

2. Objective
Use long division and synthetic division to divide polynomials.

3. Vocabulary
synthetic division

4. Polynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.

5. Example 1: Using Long Division to Divide a Polynomial
Divide using long division.
(–y2 + 2y3 + 25) ÷ (y – 3)

6. Example 2
Divide using long division.
(15x2 + 8x – 12) ÷ (3x + 1)

7. Example 3
Divide using long division.
(x2 + 5x – 28) ÷ (x – 3)

8. Example 4
Divide using long division.
1. Divide by using long division. (8x3 + 6x2 + 7) ÷ (x + 2)

9. Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form (x – a).

11. Example 1: Using Synthetic Division to Divide by a Linear Binomial
Divide using synthetic division. 1 3
(3x2 + 9x – 2) ÷ (x – )

12. Example 2: Using Synthetic Division to Divide by a Linear Binomial
Divide using synthetic division.
(3x4 – x3 + 5x – 1) ÷ (x + 2)

13. Example 3
Divide using synthetic division.
(6x2 – 5x – 6) ÷ (x + 3)

14. Example 4
Divide using synthetic division.
(x2 – 3x – 18) ÷ (x – 6)

15. 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.

16. Example 5: Using Synthetic Substitution
Use synthetic substitution to evaluate the polynomial for the given value.
P(x) = 2x3 + 5x2 – x + 7 for x = 2.

17. Example 6: Using Synthetic Substitution
Use synthetic substitution to evaluate the polynomial for the given value. 1 3
P(x) = 6x4 – 25x3 – 3x + 5 for x = – .

18. Example 7
Use synthetic substitution to evaluate the polynomial for the given value.
P(x) = x3 + 3x2 + 4 for x = –3.

19. Example 8
Use synthetic substitution to evaluate the polynomial for the given value. 1 5
P(x) = 5x2 + 9x + 3 for x = .

20. Example 9: Geometry Application
Write an expression that represents the area of the top face of a rectangular prism when the height is x + 2 and the volume of the prism is x3 – x2 – 6x.

21. Example 10
Write an expression for the length of a rectangle with width y – 9 and area y2 – 14y + 45.