High School Math Solutions – Polynomials Calculator, Dividing Polynomials (Long Division)

Last post, we talked dividing polynomials using factoring and splitting up the fraction. In this post, we will talk about another method for dividing polynomials, long division. Long division with polynomials is similar to the basic numerical long division, except we are dividing variables. This is where it gets tricky. I will talk about the steps to dividing polynomials using long division to help make the process easier and go into detail.

Steps for polynomial long division:

1.  Organize each polynomial by higher order

• We want to make sure that each polynomial is written in order of the variable with the highest exponent to the variable with the lowest exponent

• you can skip this step if they are already in high order

2.  Set up in long division form

• The denominator becomes the divisor and the numerator becomes the dividend

3.  Write 0 as the coefficient for missing terms in the dividend

• Since we’ve put in order the terms based on their exponent, we can see which terms are missing (i.e. x^4+x^2 we can see we are missing an x^3 term so we will add that in to its proper spot and make the coefficient 0) 

• This will help you with step 6, so you don’t subtract the wrong terms 

• You can skip this step if there are no missing terms

4.   Divide the first term of the dividend (numerator) by the first term of the divisor (denominator)

• This is allows us to see what we need to multiply the divisor by to get rid of the first term of the dividend

5.   Multiply the divisor by that term

•  Write the term down on top of line where the term that is getting eliminated is 

6.  Subtract this from the dividend 

• This gives you a new polynomial to work with

7.  Repeat steps 4-6 until you get a remainder 

• When you repeat step 4, move onto the newest first term from step 6

8.  Put the remainder over the divisor to create a fraction and add it to the new polynomial

This may seem a bit confusing, so we will go through two examples step by step to understand better how to solve these problems.

First example (click here):
                                                         \frac{(x^4+6x^2+2)}{(x^2+5)}

1.  Organize each polynomial by high order
     We can skip this step because the polynomials are already in high order
2.  Set up in long division form

3.  Write 0 as the coefficient for missing terms in the dividend

4.  Divide the first term of the dividend (numerator) by the first term of the divisor (denominator)
                                                   \frac{x^4}{x^2}= x^2

5.  Multiply the divisor by that term
     x^2∙(x^2+5)=x^4+5x^2

6.  Subtract this from the dividend

7.  Repeat step 4-6 until you get a remainder
     \frac{x^2}{x^2} =1

8.  Put the remainder over the divisor to create a fraction and add it to the new polynomial
                                                              x^2+1+\frac{(-3)}{(x^2+5)}

 Last example (click here):
                                                             \frac{(2x^2-18+5x)}{(x+4)}

1.  Organize each polynomial by high order
                                                              \frac{(2x^2+5x-18)}{(x+4)}

2.  Set up in long division form

3.  Write 0 as the coefficient for missing terms in the dividend
     We can skip this step because there are no missing terms.

4.  Divide the first term of the dividend (numerator) by the first term of the divisor (denominator)
                                                                \frac{2x^2}{x}=2x

5.  Multiply the divisor by that term
     2x(x+4)=2x^2+8x

6.  Subtract this from the dividend

7.  Repeat steps 4-6 until you get the remainder
     \frac{(-3x)}{x}=-3

8.  Put the remainder over the divisor to create a fraction and add it to the new polynomial
                                                                  2x-3+\frac{(-6)}{(x+4)}

Dividing polynomials using long division is very tricky. It is so easy to skip an exponent, have an algebraic error, and forget a step. This is why practicing this type of problem is so important. The only way to get better at it is to keep practicing it. Check out Symbolab’s Practice for practice problems and quizzes.

Until next time,

Leah.