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Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(9x^6 + 21x^4 + 3x^3\)[/tex], follow these steps:
1. Identify the GCF of the coefficients: Look at the numerical coefficients in the polynomial: 9, 21, and 3. The greatest common factor of these numbers is 3 because 3 is the largest number that divides each of them exactly.
2. Identify the smallest power of [tex]\(x\)[/tex]: The terms in the polynomial have the variable [tex]\(x\)[/tex] raised to different powers: [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex]. The smallest power of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
3. Factor out the GCF: Combine the GCF of the coefficients (3) with the smallest power of [tex]\(x\)[/tex] ([tex]\(x^3\)[/tex]) to determine the overall GCF of the polynomial, which is [tex]\(3x^3\)[/tex].
4. Divide each term by the GCF:
- [tex]\(9x^6 \div 3x^3 = 3x^3\)[/tex]
- [tex]\(21x^4 \div 3x^3 = 7x\)[/tex]
- [tex]\(3x^3 \div 3x^3 = 1\)[/tex]
5. Write the factored form: Now, express the original polynomial as a product of the GCF and the resulting polynomial:
[tex]\[
9x^6 + 21x^4 + 3x^3 = 3x^3(3x^3 + 7x + 1)
\][/tex]
So, the polynomial factored by its GCF is [tex]\(3x^3(3x^3 + 7x + 1)\)[/tex].
1. Identify the GCF of the coefficients: Look at the numerical coefficients in the polynomial: 9, 21, and 3. The greatest common factor of these numbers is 3 because 3 is the largest number that divides each of them exactly.
2. Identify the smallest power of [tex]\(x\)[/tex]: The terms in the polynomial have the variable [tex]\(x\)[/tex] raised to different powers: [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex]. The smallest power of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
3. Factor out the GCF: Combine the GCF of the coefficients (3) with the smallest power of [tex]\(x\)[/tex] ([tex]\(x^3\)[/tex]) to determine the overall GCF of the polynomial, which is [tex]\(3x^3\)[/tex].
4. Divide each term by the GCF:
- [tex]\(9x^6 \div 3x^3 = 3x^3\)[/tex]
- [tex]\(21x^4 \div 3x^3 = 7x\)[/tex]
- [tex]\(3x^3 \div 3x^3 = 1\)[/tex]
5. Write the factored form: Now, express the original polynomial as a product of the GCF and the resulting polynomial:
[tex]\[
9x^6 + 21x^4 + 3x^3 = 3x^3(3x^3 + 7x + 1)
\][/tex]
So, the polynomial factored by its GCF is [tex]\(3x^3(3x^3 + 7x + 1)\)[/tex].
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