Chapter 6: Polynomials and Polynomial FunctionsSection 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.

• 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.

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 wouldwhen dividing numbers.

Examples:

a) Divide x^2 + 2x - 30 by x – 5

d) (9x^3 18x^2 – x +2) ÷ (3x + 1)

Page 321: Quick check #1

Divide x^2 3x + 1 by x - 4

Example: Determine if x + 2 is a factor of each polynomial.

a)x^2 + 10x + 16

b) x^3 + 7x^2 5x - 6

Synthetic division is a shorthand method ofdividing a polynomial by a linear binomial byusing only the coefficients. For synthetic divisionto work, the polynomial must be written instandard form, using 0 as a coefficient for anymissing terms, and the divisor must be in theform (x – a).

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

Use synthetic division to divide5x^3 6x^2 + 4x -1 by x – 3

Use synthetic division to divide2x^3 + 3x^2 - 17x -30 by x + 2Then completely factor the dividend

Page 322 Quick Check #3 x^3 + 4x^2 + x- 6÷ x + 1

Use synthetic division to divideX^3 2x^2 - 5x + 6 by x + 2Then completely factor the dividend.

We know when the remainder = 0 the divisor is a factor of the polynomialin 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 = 0This is called the Remainder Theorem.

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 = 4For P(x) = x^4 5x^2+ 4x + 12

P(-4)= ?

Use remainder theorem to find the remainder forthe divisor x + 1 for p(x) = 2x^4 + 6x^3 5x^2-60

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