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Answer :
Sure! Let's factor out the largest common factor from the expression [tex]\(7x^6 - 49x^2\)[/tex].
### Step-by-Step Solution:
1. Identify the common factor in each term:
We start with the expression [tex]\(7x^6 - 49x^2\)[/tex]. Notice that both terms have a common numerical factor and a common power of [tex]\(x\)[/tex].
2. Find the greatest common factor (GCF):
- The numerical coefficients are 7 and 49. The GCF of 7 and 49 is 7 because 7 is the largest number that divides both 7 and 49 evenly.
- The terms also contain powers of [tex]\(x\)[/tex]: [tex]\(x^6\)[/tex] and [tex]\(x^2\)[/tex]. The GCF for the powers of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex], because [tex]\(x^2\)[/tex] is the highest power of [tex]\(x\)[/tex] that divides both [tex]\(x^6\)[/tex] and [tex]\(x^2\)[/tex] evenly.
Therefore, the GCF of the entire expression is [tex]\(7x^2\)[/tex].
3. Factor out the GCF:
- When we factor out [tex]\(7x^2\)[/tex] from [tex]\(7x^6\)[/tex], we divide [tex]\(7x^6\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(x^4\)[/tex].
- When we factor out [tex]\(7x^2\)[/tex] from [tex]\(-49x^2\)[/tex], we divide [tex]\(-49x^2\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(-7\)[/tex].
Thus, factoring [tex]\(7x^2\)[/tex] from [tex]\(7x^6 - 49x^2\)[/tex] gives us:
[tex]\[
7x^6 - 49x^2 = 7x^2(x^4 - 7)
\][/tex]
### Final Answer:
[tex]\[
7x^6 - 49x^2 = 7x^2(x^4 - 7)
\][/tex]
### Step-by-Step Solution:
1. Identify the common factor in each term:
We start with the expression [tex]\(7x^6 - 49x^2\)[/tex]. Notice that both terms have a common numerical factor and a common power of [tex]\(x\)[/tex].
2. Find the greatest common factor (GCF):
- The numerical coefficients are 7 and 49. The GCF of 7 and 49 is 7 because 7 is the largest number that divides both 7 and 49 evenly.
- The terms also contain powers of [tex]\(x\)[/tex]: [tex]\(x^6\)[/tex] and [tex]\(x^2\)[/tex]. The GCF for the powers of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex], because [tex]\(x^2\)[/tex] is the highest power of [tex]\(x\)[/tex] that divides both [tex]\(x^6\)[/tex] and [tex]\(x^2\)[/tex] evenly.
Therefore, the GCF of the entire expression is [tex]\(7x^2\)[/tex].
3. Factor out the GCF:
- When we factor out [tex]\(7x^2\)[/tex] from [tex]\(7x^6\)[/tex], we divide [tex]\(7x^6\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(x^4\)[/tex].
- When we factor out [tex]\(7x^2\)[/tex] from [tex]\(-49x^2\)[/tex], we divide [tex]\(-49x^2\)[/tex] by [tex]\(7x^2\)[/tex] to get [tex]\(-7\)[/tex].
Thus, factoring [tex]\(7x^2\)[/tex] from [tex]\(7x^6 - 49x^2\)[/tex] gives us:
[tex]\[
7x^6 - 49x^2 = 7x^2(x^4 - 7)
\][/tex]
### Final Answer:
[tex]\[
7x^6 - 49x^2 = 7x^2(x^4 - 7)
\][/tex]
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