variables with exponents

Question

In Algebra 2, I am given the following problem:

 

(x⁴ - y⁴) ÷ (x + y)

 

This type of equation is called advanced polynomial division in my book. How do I solve this?

 

 

 

Answer 1

Michael is correct in his answer which he arrived at by factoring and cancelling.  I'm just wondering if you meant it to be worked out like a long division problem.  If so we have...

For long division you multiply the (x+y) each time by whatever makes it equal to the first term in the remainder and then subtract just like in regular division...

The first thing you would multiply (x+y) by is x³ to make the first term in the product equal to the first term (x⁴) in the divisor.

 

 

           x³ - x²y + xy² - y³ 

         ______________

(x+y)√ x⁴           - y⁴

          -(x⁴+x³y)                

         -------------------

               -x³y           -y⁴

             -(-x³y-x²y²)            

           --------------------

                      x²y²          -y⁴

                    -(x²y²+xy³⁾       

              ----------------------

                          -xy³  - y⁴

                        -(-xy³ - y⁴)

                     ------------------

                                 0

 

You work it line by line as a division problem until the exponent on any "x" terms left are smaller than that of the polynomial being divided into the other.  In this case we have no remainder so it actually divided evenly and our answer is:

 

x³ -x²y + xy² - y³

which is the same answer you get if you multiply out Michael's answer of (x² + y²)(x - y)

 

 

Answer 2

The term  (x⁴ - y⁴) is a difference of perfect squares.

 

 

[(x² + y²)(x² - y²)] / (x + y)

 

 

The term  (x² - y²) is a difference of perfect squares.

 

[(x² + y²)(x + y)(x - y)] / (x + y)

 

 

Cancelling terms, we obtain

 

(x² + y²)(x - y)