QuestionReduce the rational expression to lowest terms. Identify all numbers that $\frac{4 t-20}{t^{2}-25}$

Studdy Solution

STEP 1

What is this asking? We need to simplify a fraction with variables by factoring and then canceling out any common factors. Watch out! Remember that we can only cancel factors, not terms!
Also, keep an eye out for values of $t$ that would make the denominator zero, even after simplification.

STEP 2

1. Factor the numerator
2. Factor the denominator
3. Simplify the expression

STEP 3

Alright, let's factor the numerator!
We've got $4t - 20$ .
We see that both terms are divisible by 4.
So, we can pull out a 4, which gives us $4(t - 5)$ .
Boom!

STEP 4

Now, let's tackle the denominator: $t^2 - 25$ .
This is a difference of squares, which means it fits the pattern $a^2 - b^2 = (a + b)(a - b)$ .

STEP 5

In our case, $a = t$ and $b = 5$ because $t^2$ is $t$ squared, and $25$ is $5$ squared.
So, $t^2 - 25$ factors into $(t + 5)(t - 5)$ .
Awesome!

STEP 6

Now, let's put it all together.
Our original expression was $\frac{4t - 20}{t^2 - 25}$ .
We factored the numerator to get $4(t - 5)$ and the denominator to get $(t + 5)(t - 5)$ .

STEP 7

So, our fraction becomes $\frac{4(t - 5)}{(t + 5)(t - 5)}$ .
Notice that we have a common factor of $(t - 5)$ in both the numerator and denominator.
Since $(t-5)$ divided by $(t-5)$ is one, we can simplify by multiplying by one, removing the $(t-5)$ factor from both top and bottom.

STEP 8

This leaves us with $\frac{4}{t + 5}$ !
But wait!
We need to remember those values of $t$ that would make the original denominator zero.
That would be when $t^2 - 25 = 0$ , which means $t = 5$ or $t = -5$ .
So, our simplified expression is valid as long as $t \ne 5$ and $t \ne -5$ .

STEP 9

The simplified expression is $\frac{4}{t + 5}$ , where $t \ne 5$ and $t \ne -5$ .

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