Dividing polynomials involves a process known as algebraic manipulation. We can view our dividend as the numerator of a fraction and our divisor as the denominator.

Consider the expression: \dfrac{3x^{2} + 7x - 6}{3x-2}. The dividend is 3x^{2} + 7x - 6 and the divisor is 3x-2. We can use algebra tiles to model the division. We will create a rectangle to represent the dividend, and the side lengths of the rectangle will represent its factors.

A visual model for division.There are 3 positive variable tiles and two negative unit tiles shown. Ask your teacher for more information.

\text{ }

We already know one of the factors is the divisor so we can make one of the side lengths 3x-2.

A visual model for division. Algebra tiles for 3x-2 is shown above the the line. Below the line are algebra tiles for 3x^2 +7x-6. Ask your teacher for more information.

Next we fill in the rectangle with 3 of the x^{2} tiles, 7 of the positive x tiles, and 6 of the -1 tiles to represent the dividend 3x^{2}+7x-6, making sure to line up tiles with equal lengths.

Notice that there are some empty spaces that we need to fill in with zero pairs.

A visual model for division. Algebra tiles for 3x-2 is shown above the the line. Below the line are algebra tiles for 3x^2 +7x-6. Blank spaces beside the x^2 tiles are filled with two negative variable tiles. The space below the seven positive variable tiles are filled with two positive variable tiles. On the left is another line and across it are variable tiles for x+3. Ask your teacher for more information.

The empty spaces are the right size for x tiles. Since we need to add zero pairs so that we don't change the value of the expression we will fill 2 of the spaces with positive x tiles and the other 2 spaces with -x tiles.

Now we can see that the length of the left side of the rectangle is x+3. This means that \dfrac{3x^{2} + 7x - 6}{3x-2}=x+3.

We can approach this algebraically by following these steps:

Completely factor both the numerator and denominator.
Identify all common factors that are present in both the numerator and the denominator. These could be monomial or binomial factors.
Divide out all common factors from the numerator and denominator.
Simplify the resulting expression.

Examples

Example 1

Factor and simplify: \dfrac{2x^{2}+10x-100}{2x+20}

Worked Solution

Create a strategy

We need to factor both the numerator and denominator by first factoring the greatest common factor of the terms in each expression.

Apply the idea

\displaystyle \dfrac{2x^{2}+10x-100}{2x+20}	\displaystyle =	\displaystyle \dfrac{2\left(x+10\right)\left(x-5\right)}{2\left(x+10\right)}	Factor the numerator and denominator
	\displaystyle =	\displaystyle \dfrac{\cancel{2}\cancel{\left(x+10\right)}\left(x-5\right)}{\cancel{2}\cancel{\left(x+10\right)}}	Divide out the common factors
	\displaystyle =	\displaystyle x-5	Simplify common factors to 1

Reflect and check

When we have multiple common factors, it is as simple as dividing out each factor separately.

Example 2

Factor and simplify: \dfrac{a^{2}-81}{9-a}

Worked Solution

Create a strategy

We can use the formula for factoring a difference of two squares: A^{2}-B^{2}=\left(A + B\right)\left(A - B\right)

Apply the idea

\displaystyle \dfrac{a^{2}-81}{9-a}	\displaystyle =	\displaystyle \dfrac{\left(a+9\right)\left(a-9\right)}{-\left(a-9\right)}	Factor the numerator and denominator
	\displaystyle =	\displaystyle \dfrac{\left(a+9\right)\cancel{\left(a-9\right)}}{-\cancel{\left(a-9\right)}}	Divide out the common factors and simplify to 1
	\displaystyle =	\displaystyle - \left(a + 9\right)	Simplify common factors to 1

Example 3

Factor and simplify: \dfrac{8x^{2} - 36x - 20}{\left(2x+1\right)\left(2x-1\right)}

Worked Solution

Create a strategy

Start by factoring the numerator. Then, we can divide out common factors.

Apply the idea

\displaystyle \dfrac{8x^{2} - 36x - 20}{\left(2x+1\right)\left(2x-1\right)}	\displaystyle =	\displaystyle \dfrac{4\left(x-5\right)\left(2x+1\right)}{\left(2x+1\right)\left(2x-1\right)}	Factor numerator
	\displaystyle =	\displaystyle \dfrac{4\left(x-5\right)\cancel{\left(2x+1\right)}}{\cancel{\left(2x+1\right)}\left(2x-1\right)}	Divide out common factors
	\displaystyle =	\displaystyle \dfrac{4\left(x-5\right)}{2x-1}	Simplify common factors to 1

Idea summary

To divide polynomials:

Completely factor the numerator and denominator
Divide out all common factors between the numerator and denominator
Simplify the resulting expression (if necessary)

Outcomes

A.EO.2

The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2d

Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor.

6.08 Divide polynomials

Ideas

Divide polynomials

Examples

Example 1

Example 2

Example 3

Outcomes

A.EO.2

A.EO.2d

What is Mathspace

About Mathspace