SOLUTION: Simplify the rational expression. State any restrictions on the variable. Much appreciated!! t2+2t-24/t2-36

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Simplify the rational expression. State any restrictions on the variable. Much appreciated!! t2+2t-24/t2-36      Log On


   

Question 718682: Simplify the rational expression. State any restrictions on the variable.
Much appreciated!!
t2+2t-24/t2-36
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First of all, please use the "^" character to indicate exponents (just like on a calculator) and put numerators and denominators in parentheses. Assuming t2 means t^2, what you posted meant:
t%5E2%2B2t-24%2Ft%5E2-36
But I suspect what you intended was:
%28t%5E2%2B2t-24%29%2F%28t%5E2-36%29
Unclear problems are less likely to get a response. So it is to your advantage to take the time to post correctly.

%28t%5E2%2B2t-24%29%2F%28t%5E2-36%29
To do both parts of this problem, simplify and find restrictions, we need to see the fraction with numerator and denominator in factored form. Factoring we get:
%28%28t%2B6%29%28t-4%29%29%2F%28%28t%2B6%29%28t-6%29%29

Any time you have a variable in a denominator, you must make sure that the variable never has a value that would turn that denominator into a zero (since we may never divide by zero!). Our denominator is (t+6)(t-6). Perhaps you can already see what values for t that would make this denominator zero. If not, then set the denominator to zero:
(t+6)(t-6) = 0
and solve. (We could have used the original denominator, t%5E2-6+=+0, but we would have factored that to solve it anyway.) Using the Zero Product Property:
t+6 = 0 or t-6 = 0
Then solve these. (I'll leave that up to you to finish.) Remember, these are values we must not allow t to have!

Now we can simplify. Looking at:
%28%28t%2B6%29%28t-4%29%29%2F%28%28t%2B6%29%28t-6%29%29
we can see that there are factors we can cancel:
%28cross%28%28t%2B6%29%29%28t-4%29%29%2F%28cross%28%28t%2B6%29%29%28t-6%29%29
leaving:
%28t-4%29%2F%28t-6%29
This is the simplified fraction. (Note: The t's here do not cancel! Only factors may be canceled and neither of those t's are factors of the numerator or denominator.)