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.
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?
Simplify the following expression, for $x\ne0$x≠0:
$\frac{9x^3+5x^5}{x}$9x3+5x5x
Simplify the following expression, for $x\ne0$x≠0:
$\frac{48x^4+12x^3-30x^2-42x}{6x}$48x4+12x3−30x2−42x6x
The triangle shown below has an area of $13n^3+11n^2+29n$13n3+11n2+29n.
Find a polynomial expression for its height.
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.
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 |
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 |
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) |
Factor and simplify $\frac{a^2-81}{9-a}$a2−819−a.
Factor and simplify:
$\frac{2x^2+10x-100}{2x+20}$2x2+10x−1002x+20
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
Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property