# 7.08 Dividing polynomials

Lesson

## Dividing by a monomial

Dividing a polynomial by a monomial, we must break the polynomial (dividend/numerator) into the product of its factors (Mixed Review: Factoring) and then simplify common factors from the dividend/numerator and divisor/denominator.

#### Worked example

##### Question 1

Simplify $\frac{4x^3+6x^2-8x}{2x}$4x3+6x28x2x

Think: We need to break the numerator into a product of its factors by using GCF and then simplifying.

Do

 $\frac{4x^3+6x^2-8x}{2x}$4x3+6x2−8x2x​ $=$= $\frac{2x\left(2x^2+3x-4\right)}{2x}$2x(2x2+3x−4)2x​ $=$= $1\left(2x^2+3x-4\right)$1(2x2+3x−4) $=$= $2x^2+3x-4$2x2+3x−4

Reflect: The dividend had degree $3$3 and the divisor had degree $1$1, so the quotient has degree $2$2. When dividing a polynomial by a monomial, are the laws of exponents important to know?

#### Practice questions

##### Question 2

Simplify the following expression, for $x\ne0$x0:

$\frac{9x^3+5x^5}{x}$9x3+5x5x

##### Question 3

Simplify the following expression, for $x\ne0$x0:

$\frac{48x^4+12x^3-30x^2-42x}{6x}$48x4+12x330x242x6x

##### Question 4

The triangle shown below has an area of $13n^3+11n^2+29n$13n3+11n2+29n.

Find a polynomial expression for its height.

## Dividing by a binomial

When we are dividing by a binomial, we will learn many different strategies in our mathematics education. We will start with the simplest scenario where the denominator is linear and the numerator is a quadratic that factors fully to allow division.

#### Worked examples

##### Question 5

Simplify $\frac{\left(3x+2\right)\left(4x+1\right)}{4x+1}$(3x+2)(4x+1)4x+1

Think: We have previously learned that $\frac{a}{a}=1$aa=1, so when we have the same term in the numerator and denominator they will cancel out.

Do:

 $\frac{\left(3x+2\right)\left(4x+1\right)}{4x+1}$(3x+2)(4x+1)4x+1​ $=$= $\left(3x+2\right)\times\frac{4x+1}{4x+1}$(3x+2)×4x+14x+1​ $=$= $\left(3x+2\right)\times1$(3x+2)×1 $=$= $3x+2$3x+2

##### Question 6

Simplify $\frac{x^2+5x-6}{x-1}$x2+5x6x1

Think: In order to divide nicely, we should try to factor the numerator to see if we can cancel out a factor. To factor $x^2+5x-6$x2+5x6  (Mixed Review: Factoring), we are looking for two number that have a sum of $5$5 and a product of $-6$6.

Do: The two number that have a sum of $5$5 and a product of $-6$6 are $6$6 and $-1$1. So we can factor the numerator and proceed to simplify.

 $\frac{x^2+5x-6}{x-1}$x2+5x−6x−1​ $=$= $\frac{\left(x+6\right)\left(x-1\right)}{x-1}$(x+6)(x−1)x−1​ $=$= $\left(x+6\right)\times\frac{x-1}{x-1}$(x+6)×x−1x−1​ $=$= $\left(x+6\right)\times1$(x+6)×1 $=$= $x+6$x+6

#### Question 7

Simplify $\frac{4x^2-24x+36}{2x-6}$4x224x+362x6

Think: Always look for GCF before attempting to factor using any other method. The numerator has a GCF of $4$4 and a GCF of $2$2 is in the denominator. Next, try to factor the numerator to see if we can cancel out a factor  (Mixed Review: Factoring).

Do: Take out the GCF

 $\frac{4x^2-24x+36}{2x-6}$4x2−24x+362x−6​ $=$= $\frac{4\left(x^2-6x+9\right)}{2\left(x-3\right)}$4(x2−6x+9)2(x−3)​ $=$= $\frac{4}{2}\frac{x^2-6x+9}{x-3}$42​x2−6x+9x−3​ $=$= $\frac{2\left(x^2-6x+9\right)}{x-3}$2(x2−6x+9)x−3​

Factor the trinomial

 $\frac{4x^2-24x+36}{2x-6}$4x2−24x+362x−6​ $=$= $\frac{2\left(x-3\right)\left(x-3\right)}{x-3}$2(x−3)(x−3)x−3​ $=$= $2\frac{x-3}{x-3}\left(x-3\right)$2x−3x−3​(x−3) $=$= $2\left(x-3\right)$2(x−3)

#### Practice questions

##### Question 8

Factor and simplify $\frac{a^2-81}{9-a}$a2819a.

##### Question 9

Factor and simplify:

$\frac{2x^2+10x-100}{2x+20}$2x2+10x1002x+20

### Outcomes

#### A1.10.C

Determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend

#### A1.10.D

Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property