Math / Algebra

QuestionFactor completely: $4 x^{2}\left(x^{2}+5\right)-\left(x^{2}+5\right)$
Answer Attempt 1 out of 3

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression as a product of simpler expressions. Watch out! Don't forget to factor completely!

STEP 2

1. Find the greatest common factor.
2. Factor it out.
3. Factor the remaining quadratic.

STEP 3

Look closely at our expression: $4x^2(x^2 + 5) - (x^2 + 5)$ .
Notice that both terms have the common factor $(x^2 + 5)$ !
It's like they're holding hands!

STEP 4

Let's pull out that common factor, $(x^2 + 5)$ , from both terms.
Remember, when we factor something out, we're essentially dividing each term by that factor.
So, we get: $(x^2 + 5)(4x^2 - 1)$ .
See how we divided the first term, $4x^2(x^2 + 5)$ , by $(x^2 + 5)$ to get $4x^2$ ?
And when we divide the second term, $(x^2 + 5)$ , by itself, we're left with $1$ , but since it was being subtracted, we have $-1$ .

STEP 5

Now, let's look at what's left: $(4x^2 - 1)$ .
This is a difference of squares!
Remember, a difference of squares looks like $a^2 - b^2$ , and it can be factored into $(a + b)(a - b)$ .

STEP 6

In our case, $4x^2$ is like $a^2$ and $1$ is like $b^2$ .
So, $a$ is $2x$ (since $(2x)^2 = 4x^2$ ) and $b$ is $1$ (since $1^2 = 1$ ).

STEP 7

Now, we can rewrite $(4x^2 - 1)$ as $(2x + 1)(2x - 1)$ .
Awesome!

STEP 8

So, our completely factored expression is $(x^2 + 5)(2x + 1)(2x - 1)$ !
We've broken it down as much as possible.

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QuestionFactor completely: $4 x^{2}\left(x^{2}+5\right)-\left(x^{2}+5\right)$
Answer Attempt 1 out of 3

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression as a product of simpler expressions. Watch out! Don't forget to factor completely!

STEP 2

1. Find the greatest common factor.
2. Factor it out.
3. Factor the remaining quadratic.

STEP 3

Look closely at our expression: $4x^2(x^2 + 5) - (x^2 + 5)$ .
Notice that both terms have the common factor $(x^2 + 5)$ !
It's like they're holding hands!

STEP 4

STEP 5

Now, let's look at what's left: $(4x^2 - 1)$ .
This is a difference of squares!
Remember, a difference of squares looks like $a^2 - b^2$ , and it can be factored into $(a + b)(a - b)$ .

STEP 6

In our case, $4x^2$ is like $a^2$ and $1$ is like $b^2$ .
So, $a$ is $2x$ (since $(2x)^2 = 4x^2$ ) and $b$ is $1$ (since $1^2 = 1$ ).

STEP 7

Now, we can rewrite $(4x^2 - 1)$ as $(2x + 1)(2x - 1)$ .
Awesome!

STEP 8

So, our completely factored expression is $(x^2 + 5)(2x + 1)(2x - 1)$ !
We've broken it down as much as possible.

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QuestionFactor completely: 4x2(x2+5)−(x2+5)4 x^{2}\left(x^{2}+5\right)-\left(x^{2}+5\right)4x2(x2+5)−(x2+5)Answer Attempt 1 out of 3

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression as a product of simpler expressions. Watch out! Don't forget to factor *completely*!

STEP 2

1. Find the greatest common factor.2. Factor it out.3. Factor the remaining quadratic.

STEP 3

Look closely at our expression: 4x2(x2+5)−(x2+5)4x^2(x^2 + 5) - (x^2 + 5)4x2(x2+5)−(x2+5).Notice that both terms have the common factor (x2+5)(x^2 + 5)(x2+5)!It's like they're holding hands!

STEP 4

STEP 5

Now, let's look at what's left: (4x2−1)(4x^2 - 1)(4x2−1).This is a *difference of squares*!Remember, a difference of squares looks like a2−b2a^2 - b^2a2−b2, and it can be factored into (a+b)(a−b)(a + b)(a - b)(a+b)(a−b).

STEP 6

In our case, 4x24x^24x2 is like a2a^2a2 and 111 is like b2b^2b2.So, aaa is 2x2x2x (since (2x)2=4x2(2x)^2 = 4x^2(2x)2=4x2) and bbb is 111 (since 12=11^2 = 112=1).

STEP 7

Now, we can rewrite (4x2−1)(4x^2 - 1)(4x2−1) as (2x+1)(2x−1)(2x + 1)(2x - 1)(2x+1)(2x−1).Awesome!

STEP 8

So, our *completely* factored expression is (x2+5)(2x+1)(2x−1)(x^2 + 5)(2x + 1)(2x - 1)(x2+5)(2x+1)(2x−1)!We've broken it down as much as possible.

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QuestionFactor completely: $4 x^{2}\left(x^{2}+5\right)-\left(x^{2}+5\right)$
Answer Attempt 1 out of 3

What is this asking? We need to rewrite this expression as a product of simpler expressions. Watch out! Don't forget to factor completely!

1. Find the greatest common factor.
2. Factor it out.
3. Factor the remaining quadratic.

Look closely at our expression: $4x^2(x^2 + 5) - (x^2 + 5)$ .
Notice that both terms have the common factor $(x^2 + 5)$ !
It's like they're holding hands!

Now, let's look at what's left: $(4x^2 - 1)$ .
This is a difference of squares!
Remember, a difference of squares looks like $a^2 - b^2$ , and it can be factored into $(a + b)(a - b)$ .

In our case, $4x^2$ is like $a^2$ and $1$ is like $b^2$ .
So, $a$ is $2x$ (since $(2x)^2 = 4x^2$ ) and $b$ is $1$ (since $1^2 = 1$ ).

Now, we can rewrite $(4x^2 - 1)$ as $(2x + 1)(2x - 1)$ .
Awesome!

So, our completely factored expression is $(x^2 + 5)(2x + 1)(2x - 1)$ !
We've broken it down as much as possible.