1. Chapter 6: Polynomials and Polynomial Functions Section 6
Chapter 6: Polynomials and Polynomial Functions Section 6.3: Dividing Polynomials
Content Objectives: Students will demonstrate application of polynomial division by solving problems using long and synthetic division.
Language Objectives: Students will demonstrate the understanding of how to write rational expressions in polynomial form and fraction form.

2. In arithmetic long division, you follow these steps: divide, multiply, subtract, and bring down.
Follow these same steps to use long division to divide polynomials.
Polynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.

3. Example 1: Using Long Division to Divide a Polynomial
(–y^2 + 2y^3 + 25) ÷ (y – 3)
Step 1: Write the dividend in standard form, including terms with a coefficient of 0.
2y^3 – y^2 + 0y + 25
Step 2: Write division in the same way you would
when dividing numbers.

4. Examples:
Divide x^2 + 2x by x – 5
d) (9x^3 18x^2 – x +2) ÷ (3x + 1)

5. Page 321: Quick check #1
Divide x^2 3x + 1 by x - 4

6. Example: Determine if x + 2 is a factor of each polynomial.
a)x^2 + 10x + 16
b) x^3 + 7x^2 5x - 6

7. 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 as a coefficient for any
missing terms, and the divisor must be in the
form (x – a).

9. Answer is always one degree less than the dividend due to dividing by a linear term.

10. Use synthetic division to divide
5x^3 6x^2 + 4x -1 by x – 3
2x^3 + 3x^2 - 17x -30 by x + 2
Then completely factor the dividend
Page 322 Quick Check #3 x^3 + 4x^2 + x- 6
÷ x + 1

11. Use synthetic division to divide
X^3 2x^2 - 5x + 6 by x + 2
Then completely factor the dividend.
We know when the remainder = 0 the divisor is a factor of the polynomial
in the dividend.
X + 2 is a factor of x^3 2x^2 - 5x+ 6
If we evaluate f(a) or f(-2)
= (-2)^3-2(-2)^2-5(-2)+6 = 0
This is called the Remainder Theorem.

12. Theorem: If a polynomial, p(x), is divided by x a, where "a" is a constant, then the remainder is p(a).
Find the remainder if a = 4
For P(x) = x^4 5x^2+ 4x + 12
P(-4)= ?
Use remainder theorem to find the remainder for
the divisor x + 1 for p(x) = 2x^4 + 6x^3 5x^2-60

13. Text Assignment: Pg. 1-23, 27-33, 37-55 [odds]