QuestionQuestion
Factor completely: $x^{2}(x+10)-(x+10)$
Answer Attempt iout of 3

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression $x^{2}(x+10)-(x+10)$ as a product of simpler terms. Watch out! Don't forget to factor completely!
Sometimes, there are multiple steps to factoring.

STEP 2

1. Factor out the common term
2. Factor the difference of squares

STEP 3

Notice that both terms in the expression, $x^{2}(x+10)$ and $(x+10)$ , have a common factor of $(x+10)$ .
That's awesome!
This means we can factor it out!

STEP 4

Let's pull that $(x+10)$ out to the front.
This gives us $(x+10)(x^2 - 1).$ Remember, when we factor something out, we're essentially dividing each term by that factor.
In this case, we're dividing $x^{2}(x+10)$ by $(x+10)$ , which leaves us with $x^2$ , and we're dividing $(x+10)$ by itself, which leaves us with $1$ .
That's why we get $x^2 - 1$ inside the second set of parentheses.

STEP 5

Now, look closely at what's left inside the second parentheses: $x^2 - 1$ .
This is a difference of squares!
We can rewrite $1$ as $1^2$ , so we have $x^2 - 1^2$ .
Remember, the difference of squares follows the pattern $a^2 - b^2 = (a+b)(a-b)$ .

STEP 6

In our case, $a = x$ and $b = 1$ .
So, $x^2 - 1 = (x+1)(x-1)$ .
This is super important because it allows us to factor even further!

STEP 7

Now, let's put everything together.
We have $(x+10)$ from our first factoring step and $(x+1)(x-1)$ from our second step.
Multiplying these together gives us the completely factored form: $(x+10)(x+1)(x-1).$

STEP 8

The completely factored form of $x^{2}(x+10)-(x+10)$ is $(x+10)(x+1)(x-1)$ .

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