Dividing Polynomials

There are two different methods of dividing polynomials. First, we can rewrite the polynomials as rational expressions and look for common factors. Alternatively, we can use polynomial long division. Finding common factors is usually the easier method, so we try that first and resort to long division only when we need to. This article will explore both methodologies, so let's get started!

Dividing polynomials using common factors

The best way to illustrate how to divide polynomials is to look at a practice problem, so let's divide

$\left(15p^{2}q+25p{q}^2-10pq\right)\text{by}5pq$

The first step is rewriting the problem as a fraction:

$\frac{\left(15p^{2}q+25p{q}^2-10pq\right)}{5pq}$

Next, we can factor out a 5pq from the numerator since it goes into all three terms evenly:

$\frac{5pq\times \left(3p+5q-2\right)}{5pq}$

We now have a 5pq in both the numerator and denominator, meaning they cancel each other out. That gives us a final answer of

$3p+5q-2$

It isn't always easy to spot common factors, but with practice, you should get the hang of it.

Dividing polynomials using long division

Again, the best way to approach this is to look at a sample problem. Let's say we want to divide

$\left(m^2+4m-12\right)\text{by}\left(m+3\right)$ .

First, we want to change the problem into a fraction:

$\frac{m^2+4m-12}{m+3}$

We could factor the numerator into $\left(m+6\right)\left(m-2\right)$ , but that doesn't help us in this case. Therefore, we're going to have to do long division. Begin by writing your long division symbol and dividing the leading terms: $m^2÷m$ . That equals m, so put an m on the far left of your long division symbol. Next, we have to multiply $$ $\left(m+3\right)$ by m and subtract the difference from our numerator:

$m\left(m+3\right)=m^2+3m$

$m^2+4m-m^2+3m=m$

We bring down the -12 as well for $m-12$ . Now, we divide the leading terms again: m / m. That equals 1, so add a $+1$ to our quotient at the top of the long division symbol. Multiplying $m+3$ by 1 gives us $m+3$ , so we subtract that from the $m-12$ to get $-15$ $\left(m-m=0,-12-3=-15\right)$ .

We cannot do any more division, so we'll include the $\left(\frac{15}{m+3}\right)$ as a remainder in our quotient. Our final answer is:

$m+1-\left(\frac{15}{m+3}\right)$

Long division involves a lot of steps and you might remember your elementary school math teacher cautioning you to take your time with it. That becomes even more important when you're dividing polynomials.

Dividing polynomials practice problems

a. $\left(\frac{18x^2+9x-3}{9x}\right)$

We have common factors to work with. The denominator will distribute, so it might help if we consider each element separately:

$\left(\frac{18x^2}{9x}+\frac{9x}{9x}-\frac{3}{9x}\right)$

Next, we cancel the common factors for the following equation:

$\left(\frac{18}{9}\times \frac{x^2}{x},+,1,-,\frac{3}{9}\times \frac{1}{x}\right)$

When we simplify that, we get a final answer of $2x+1-\frac{1}{3}x$

b. $\left(\frac{x^2+6x-7}{x+3}\right)$

We don't have any common factors to work with, so we'll have to do long division. Begin by dividing the leading coefficients:

$\frac{x^2}{x}$

The answer is x and represents the first term of our quotient.

Next, we multiply $x$ by the divisor of $x+3$ to get $x^2+3x$ and subtract it from the dividend:

$x^2+6x-7-(x^2+3x)=0+3x-7$

Don't forget to bring down the -7. Now, we divide the leading coefficients $3x$ and $3$ for an answer of 3, the second term in our quotient. Multiply 3 by our divisor of $x+3$ to get $3x+9$ , which we subtract from the divisor as shown:

$3x-7-(3x+9)=0-16$

We can't do any more division since we're out of variables, making -16 our remainder. Our final quotient can be written as $(x+3)-\frac{16}{(x+3)}$ .

Topics related to the Dividing Polynomials

Multiplying Polynomials

Degree (of a Polynomial)

Binomials

Flashcards covering the Dividing Polynomials

Algebra II Flashcards

College Algebra Flashcards

Practice tests covering the Dividing Polynomials

Algebra II Diagnostic Tests

College Algebra Diagnostic Tests

Get help with dividing polynomials

Dividing polynomials requires students to understand which of two methodologies to choose and carefully implement it, especially since both allow plenty of room for silly mistakes. If your student feels overwhelmed by long division with variables or how to identify whether a given problem has common factors, a 1-on-1 math tutor can provide practice problems of increasing difficulty to boost their self-confidence. Your student can also go back and review any older concepts that may be hindering their efforts on more advanced material. Reach out to the friendly Educational Directors at Varsity Tutors today to learn more.