SOLUTION: Factor completely: x^7y^2 - 4x^5y^2 - 21x^3y^2 I am having a difficult time factoring this trinominal. Think I am getting confused by all the x's and y's. Could you please help

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor completely: x^7y^2 - 4x^5y^2 - 21x^3y^2 I am having a difficult time factoring this trinominal. Think I am getting confused by all the x's and y's. Could you please help      Log On


   

Question 73547This question is from textbook Intermediate Algebra
: Factor completely: x^7y^2 - 4x^5y^2 - 21x^3y^2
I am having a difficult time factoring this trinominal. Think I am getting confused by all the x's and y's. Could you please help me, so I can work these types of problems correctly? This question is from textbook Intermediate Algebra

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Factor:
.
+x%5E7y%5E2+-+4x%5E5y%5E2+-+21x%5E3y%5E2
.
First look to see if there are any common factors in every term. Note that every term has
a y%5E2 in it. So pull that out and make it a multiplier of all the terms. You should
find that you have:
.
y%5E2%2A%28x%5E7+-+4x%5E5+-+21x%5E3%29
.
Look at what is left in the parentheses. All of the terms contain an x%5E3 so factor it
out and make it another multiplier of the expression remaining in the parentheses. When you
do the result is:
.
y%5E2%2Ax%5E3%2A%28x%5E4+-+4x%5E2+-+21%29
.
So far, so good. Now examine what's left in the parentheses. You may not recognize
it, but it can be factored like a quadratic. It may help you to see that if you let
A+=+x%5E2 then the expression in the parentheses would become:
.
A%5E2+-+4A+-+21 and you may not be able to see that this factors to %28A+-+7%29%2A%28A%2B3%29.
Then you can plug x%5E2+ back in for A to get the two factors of x%5E4+-+4x%5E2+-21
as being %28x%5E2+-7%29%2A%28x%5E2+%2B+3%29
.
So this is where we are at this point:
.
y%5E2%2Ax%5E3%2A%28x%5E2+-+7%29%2A%28x%5E2+%2B+3%29
.
This may be good enough for the answer you are to provide, but there is one more mathematical
trick you can use on this. The term x%5E2+-+7 is the difference of two squares.
Therefore its factors are the sum and difference of the square roots of each term in
the parentheses. So x%5E2+-+7 factors to %28x+-+sqrt%287%29%29%2A%28x+%2B+sqrt%287%29%29. This final
step completes what can be done by factoring. The result is:
.
y%5E2%2Ax%5E3%2A%28x-sqrt%287%29%29%2A%28x%2Bsqrt%287%29%29%2A%28x%5E2+%2B+3%29
.
This last step all depends on whether your teacher expects you to be working with radicals
in the answer.
.
Hope this helps.