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2.08 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+6x28x2x $=$= $\frac{2x\left(2x^2+3x-4\right)}{2x}$2x(2x2+3x4)2x
  $=$= $1\left(2x^2+3x-4\right)$1(2x2+3x4)
  $=$= $2x^2+3x-4$2x2+3x4

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 simplified polynomial expression for its height.

A triangle is drawn with a solid green line. The base of the triangle measures $n$n units, as indicated by the scale line. The vertical height of the triangle is represented by a green dashed line drawn from the apex to the base of the triangle. A small square with a blue outline is drawn at the corner where the green dashed line and base intersect indicating that it is a right triangle.

 

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+5x6x1 $=$= $\frac{\left(x+6\right)\left(x-1\right)}{x-1}$(x+6)(x1)x1
  $=$= $\left(x+6\right)\times\frac{x-1}{x-1}$(x+6)×x1x1
  $=$= $\left(x+6\right)\times1$(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}$4x224x+362x6 $=$= $\frac{4\left(x^2-6x+9\right)}{2\left(x-3\right)}$4(x26x+9)2(x3)
  $=$= $\frac{4}{2}\frac{x^2-6x+9}{x-3}$42x26x+9x3
  $=$= $\frac{2\left(x^2-6x+9\right)}{x-3}$2(x26x+9)x3

Factor the trinomial

$\frac{4x^2-24x+36}{2x-6}$4x224x+362x6 $=$= $\frac{2\left(x-3\right)\left(x-3\right)}{x-3}$2(x3)(x3)x3
  $=$= $2\frac{x-3}{x-3}\left(x-3\right)$2x3x3(x3)
  $=$= $2\left(x-3\right)$2(x3)

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

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