7.08 Dividing polynomials
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+6x2−8x2x
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 $x\ne0$x≠0:
$\frac{9x^3+5x^5}{x}$9x3+5x5x
Question 3
Simplify the following expression, for $x\ne0$x≠0:
$\frac{48x^4+12x^3-30x^2-42x}{6x}$48x4+12x3−30x2−42x6x
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.
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+5x−6x−1
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+5x−6 (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}$4x2−24x+362x−6
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}$42x2−6x+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 $\frac{a^2-81}{9-a}$a2−819−a.
Question 9
Factor and simplify:
$\frac{2x^2+10x-100}{2x+20}$2x2+10x−1002x+20