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Answer :
Sure! Let's factor the polynomial [tex]\(5x^6 - 7x^4 - 10x^2 + 14\)[/tex] by grouping. Here's a step-by-step guide to doing this:
1. Identify pairs to group:
We will first split the polynomial into two pairs:
[tex]\[
(5x^6 - 7x^4) \quad \text{and} \quad (-10x^2 + 14)
\][/tex]
2. Factor each group:
- For the first pair [tex]\(5x^6 - 7x^4\)[/tex], we can factor out the greatest common factor (GCF), which is [tex]\(x^4\)[/tex]:
[tex]\[
x^4(5x^2 - 7)
\][/tex]
- For the second pair [tex]\(-10x^2 + 14\)[/tex], we can factor out the GCF, which is [tex]\(-2\)[/tex]:
[tex]\[
-2(5x^2 - 7)
\][/tex]
3. Notice the common factor:
After factoring, we notice that both groups have a common binomial factor [tex]\((5x^2 - 7)\)[/tex].
4. Factor out the common factor:
We can factor out [tex]\((5x^2 - 7)\)[/tex] from the entire expression:
[tex]\[
(5x^2 - 7)(x^4 - 2)
\][/tex]
The expression is now factored completely by grouping:
[tex]\[
(5x^2 - 7)(x^4 - 2)
\][/tex]
And that's our factored expression!
1. Identify pairs to group:
We will first split the polynomial into two pairs:
[tex]\[
(5x^6 - 7x^4) \quad \text{and} \quad (-10x^2 + 14)
\][/tex]
2. Factor each group:
- For the first pair [tex]\(5x^6 - 7x^4\)[/tex], we can factor out the greatest common factor (GCF), which is [tex]\(x^4\)[/tex]:
[tex]\[
x^4(5x^2 - 7)
\][/tex]
- For the second pair [tex]\(-10x^2 + 14\)[/tex], we can factor out the GCF, which is [tex]\(-2\)[/tex]:
[tex]\[
-2(5x^2 - 7)
\][/tex]
3. Notice the common factor:
After factoring, we notice that both groups have a common binomial factor [tex]\((5x^2 - 7)\)[/tex].
4. Factor out the common factor:
We can factor out [tex]\((5x^2 - 7)\)[/tex] from the entire expression:
[tex]\[
(5x^2 - 7)(x^4 - 2)
\][/tex]
The expression is now factored completely by grouping:
[tex]\[
(5x^2 - 7)(x^4 - 2)
\][/tex]
And that's our factored expression!
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