 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