Answer :
To factor the polynomial [tex]\(9x^7 + 33x^3\)[/tex], let's go through a step-by-step process:
1. Identify the greatest common factor (GCF):
Look for the greatest common factor in the terms [tex]\(9x^7\)[/tex] and [tex]\(33x^3\)[/tex].
- The coefficients 9 and 33 have a greatest common factor of 3.
- The variable parts [tex]\(x^7\)[/tex] and [tex]\(x^3\)[/tex] have a common factor of [tex]\(x^3\)[/tex].
Therefore, the GCF of the polynomial is [tex]\(3x^3\)[/tex].
2. Factor out the GCF:
Divide each term of the polynomial by the GCF [tex]\(3x^3\)[/tex].
- For the first term: [tex]\(\frac{9x^7}{3x^3} = 3x^4\)[/tex].
- For the second term: [tex]\(\frac{33x^3}{3x^3} = 11\)[/tex].
By factoring out [tex]\(3x^3\)[/tex], the polynomial becomes:
[tex]\[
3x^3(3x^4 + 11)
\][/tex]
3. Check your work:
After factoring, you can verify the factorization by expanding the expression:
- Distribute [tex]\(3x^3\)[/tex] to each term inside the parentheses:
[tex]\[
3x^3 \times 3x^4 = 9x^7
\][/tex]
[tex]\[
3x^3 \times 11 = 33x^3
\][/tex]
Combining these, you get back to the original polynomial [tex]\(9x^7 + 33x^3\)[/tex]. This confirms the factorization is correct.
Therefore, the correctly factored form of the polynomial [tex]\(9x^7 + 33x^3\)[/tex] is:
[tex]\[
3x^3(3x^4 + 11)
\][/tex]
This matches the option: [tex]\(3x^3\left(3x^4 + 11\right)\)[/tex].
1. Identify the greatest common factor (GCF):
Look for the greatest common factor in the terms [tex]\(9x^7\)[/tex] and [tex]\(33x^3\)[/tex].
- The coefficients 9 and 33 have a greatest common factor of 3.
- The variable parts [tex]\(x^7\)[/tex] and [tex]\(x^3\)[/tex] have a common factor of [tex]\(x^3\)[/tex].
Therefore, the GCF of the polynomial is [tex]\(3x^3\)[/tex].
2. Factor out the GCF:
Divide each term of the polynomial by the GCF [tex]\(3x^3\)[/tex].
- For the first term: [tex]\(\frac{9x^7}{3x^3} = 3x^4\)[/tex].
- For the second term: [tex]\(\frac{33x^3}{3x^3} = 11\)[/tex].
By factoring out [tex]\(3x^3\)[/tex], the polynomial becomes:
[tex]\[
3x^3(3x^4 + 11)
\][/tex]
3. Check your work:
After factoring, you can verify the factorization by expanding the expression:
- Distribute [tex]\(3x^3\)[/tex] to each term inside the parentheses:
[tex]\[
3x^3 \times 3x^4 = 9x^7
\][/tex]
[tex]\[
3x^3 \times 11 = 33x^3
\][/tex]
Combining these, you get back to the original polynomial [tex]\(9x^7 + 33x^3\)[/tex]. This confirms the factorization is correct.
Therefore, the correctly factored form of the polynomial [tex]\(9x^7 + 33x^3\)[/tex] is:
[tex]\[
3x^3(3x^4 + 11)
\][/tex]
This matches the option: [tex]\(3x^3\left(3x^4 + 11\right)\)[/tex].
