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Answer :
To factor the expression [tex]\(9x^7 + 21x^6 + 33x^4\)[/tex], let's follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for the GCF of all the terms: [tex]\(9x^7\)[/tex], [tex]\(21x^6\)[/tex], and [tex]\(33x^4\)[/tex].
- The numerical coefficients are 9, 21, and 33. The GCF of these numbers is 3.
- The variable part has powers of [tex]\(x^7\)[/tex], [tex]\(x^6\)[/tex], and [tex]\(x^4\)[/tex]. The GCF here is [tex]\(x^4\)[/tex], since it is the lowest power of [tex]\(x\)[/tex] common in all terms.
2. Factor out the GCF:
Now we factor out [tex]\(3x^4\)[/tex] from the original expression:
[tex]\[
9x^7 + 21x^6 + 33x^4 = 3x^4(3x^3 + 7x^2 + 11)
\][/tex]
- From [tex]\(9x^7\)[/tex], once you factor out [tex]\(3x^4\)[/tex], you are left with [tex]\(3x^3\)[/tex].
- From [tex]\(21x^6\)[/tex], once you factor out [tex]\(3x^4\)[/tex], you are left with [tex]\(7x^2\)[/tex].
- From [tex]\(33x^4\)[/tex], once you factor out [tex]\(3x^4\)[/tex], you are left with 11.
3. Verify the Factored Form:
You can multiply the factors back together to verify:
[tex]\[
3x^4(3x^3 + 7x^2 + 11) = 9x^7 + 21x^6 + 33x^4
\][/tex]
Hence, the completely factored form of the expression is [tex]\(3x^4(3x^3 + 7x^2 + 11)\)[/tex].
1. Identify the Greatest Common Factor (GCF):
First, look for the GCF of all the terms: [tex]\(9x^7\)[/tex], [tex]\(21x^6\)[/tex], and [tex]\(33x^4\)[/tex].
- The numerical coefficients are 9, 21, and 33. The GCF of these numbers is 3.
- The variable part has powers of [tex]\(x^7\)[/tex], [tex]\(x^6\)[/tex], and [tex]\(x^4\)[/tex]. The GCF here is [tex]\(x^4\)[/tex], since it is the lowest power of [tex]\(x\)[/tex] common in all terms.
2. Factor out the GCF:
Now we factor out [tex]\(3x^4\)[/tex] from the original expression:
[tex]\[
9x^7 + 21x^6 + 33x^4 = 3x^4(3x^3 + 7x^2 + 11)
\][/tex]
- From [tex]\(9x^7\)[/tex], once you factor out [tex]\(3x^4\)[/tex], you are left with [tex]\(3x^3\)[/tex].
- From [tex]\(21x^6\)[/tex], once you factor out [tex]\(3x^4\)[/tex], you are left with [tex]\(7x^2\)[/tex].
- From [tex]\(33x^4\)[/tex], once you factor out [tex]\(3x^4\)[/tex], you are left with 11.
3. Verify the Factored Form:
You can multiply the factors back together to verify:
[tex]\[
3x^4(3x^3 + 7x^2 + 11) = 9x^7 + 21x^6 + 33x^4
\][/tex]
Hence, the completely factored form of the expression is [tex]\(3x^4(3x^3 + 7x^2 + 11)\)[/tex].
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