Study Guide

Basic Algebra - Dividing Polynomials

Dividing Polynomials

Dividing polynomials starts with dividing monomials, and dividing monomials boils down to reducing fractions, and reducing fractions? Pshaw, we've been doing that for eons. No big whoop. The fractions have variables now, but so what? We've got this.

The most important thing to remember is that when we divide a variable by itself, it equals 1, just like 5 ÷ 5 = 1 or x ÷ x = 1.

Dividing Monomials by Monomials

Let's look at an example. Remember, fractions are just another way to write division.

18y^3/9y

Instead of writing y^3, we can write yyy (which means y x y x y).

18yyy/9y

Now we can divide, or reduce, the coefficients and the variables. 18 / 9 = 2 and y/ y=1, just like 7/7=1.

reduce coefficients 18yyy/9y

Simplified, this looks like:

(2x1yy)/(1x1) = 2y^2/1=2y^2

Sample Problem

Divide 125x2y by 150xy2.

For simplicity, we can write this as a fraction:

125xxy/150xyy

Now let's write the variables the long way.

125xxy/150xyy

Then reduce:

reducing 125xxy/150xyy

Simplified, it look like this:

(5*1*1x)/6y = 5x/6y

Sample Problem

Divide 64abc/8abcd.

There are no variables with exponents that we need to write out, so we can go straight into reducing:

reducing 64abc/8abcd

Simplify that beast.

8*1*1*1/1*1*1*1d=8/d

Dividing Polynomials by Monomials

We may also need to divide polynomials by monomials. To do this, we need to separate the "fractions" into smaller fractions with just one term in each numerator.

(16x^2 + 24x)/(-4x^2)

We can rewrite this fraction as:

16x^2/-4x^2 + 24x/-4x^2

(Remember, when you add fractions together, you combine the numerators and keep the denominator.)

Now, let's write out the variables the long way:

(16xx)/(-4xxx)+(24x)/(-4xx)

Then reduce:

reducing 16xx/-4xx + 24x/-4xx

Simplified, it looks like this:

(4*1*1)/(-1*1*1) + (6*1)/(-1*1*1)

4/-1 + 6/-x

-4 + 6/-x

Look Out: watch your negative signs in a fraction bar. 2/-a is the same as -2/a, which is also the same as -(2/a), but it's not the same as -2/-a , which would equal  2/a .

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