Question 718682
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^2+2t-24/t^2-36}}}
But I suspect what you intended was:
{{{(t^2+2t-24)/(t^2-36)}}}
Unclear problems are less likely to get a response. So it is to your advantage to take the time to post correctly.<br>
{{{(t^2+2t-24)/(t^2-36)}}}
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:
{{{((t+6)(t-4))/((t+6)(t-6))}}}<br>
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 <u>never</u> 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^2-6 = 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 <u>not</u> allow t to have!<br>
Now we can simplify. Looking at:
{{{((t+6)(t-4))/((t+6)(t-6))}}}
we can see that there are factors we can cancel:
{{{(cross((t+6))(t-4))/(cross((t+6))(t-6))}}}
leaving:
{{{(t-4)/(t-6)}}}
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.)