Dividing Polynomials
Topic Index | Algebra Index | Regents Exam Prep Center

 

We will be examining polynomials divided by monomials and by binomials.
 

Steps for Dividing a Polynomial by a Monomial:

1. Divide each term of the polynomial by the monomial.

         a)  Divide numbers (coefficients)
         b)  Subtract exponents

Keep this in mind

* The number of terms in the polynomial equals the number of terms in the answer when dividing by a monomial.


2.
 
Remember that numbers do not cancel and disappear!  A number divided by itself is 1.  It reduces to the number 1.

3. 
Remember to write the appropriate sign in between the terms.
 
 
Example: 

 

The polynomial on the top has 3 terms and the answer has 3 terms.

Notice how the numbers (the coefficients) were divided. 

Answer:
 

Notice how the exponents were subtracted.

Notice how the last term reduced to one.

 

 

Think about it:

Dividing by a number is the same process as multiplying by the reciprocal of that number.

 

Notice how we used the reciprocal of 4x2  

Answer:
 

Now the reciprocal was distributed across the parentheses and the problem proceeds as in the example above.

 

Steps for Dividing a Polynomial by a Binomial:

1. Remember that the terms in a binomial cannot be separated from one another when reducing.  For example, in the binomial 2x + 3, the 2x can never be reduced unless the entire
expression  2x + 3 is reduced.

2.
 
Factor completely both the numerator and denominator before reducing.

3. 
Divide both the numerator and denominator by their greatest common factor.

Examples:  (see more under "Factoring" section)

1.

 

Notice that the x+1 was reduced as a "set".

2.


 
 
3.

 

 
4.


 

Tricky strategy:  Notice that the -1 was factored out of the numerator to create a binomial compatible with the one in the denominator.
2 - x = -1(x - 2)

 

If you are solving multiple choice questions, you can use the calculator to "check" your work with polynomials.
You can use a Numerical Checking process:  Numerical Process
                                  or
You can use an Equation Checking process: Equation Process