Answer :
To factor the polynomial [tex]\(9x^4 + 21x^3 + 12x^2\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients and the variable powers in the terms [tex]\(9x^4\)[/tex], [tex]\(21x^3\)[/tex], and [tex]\(12x^2\)[/tex].
- The GCF of the coefficients [tex]\(9\)[/tex], [tex]\(21\)[/tex], and [tex]\(12\)[/tex] is [tex]\(3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] in all the terms is [tex]\(x^2\)[/tex].
- Therefore, the GCF of the entire polynomial is [tex]\(3x^2\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF, [tex]\(3x^2\)[/tex]:
[tex]\[
9x^4 + 21x^3 + 12x^2 = 3x^2(3x^2 + 7x + 4)
\][/tex]
3. Factor the remaining quadratic expression, [tex]\(3x^2 + 7x + 4\)[/tex]:
- Look for two numbers that multiply to [tex]\(3 \times 4 = 12\)[/tex] and add to [tex]\(7\)[/tex]. These numbers are [tex]\(3\)[/tex] and [tex]\(4\)[/tex].
4. Rewrite the middle term ([tex]\(7x\)[/tex]) using [tex]\(3\)[/tex] and [tex]\(4\)[/tex]:
- Rewrite [tex]\(7x\)[/tex] as [tex]\(3x + 4x\)[/tex]:
[tex]\[
3x^2 + 7x + 4 = 3x^2 + 3x + 4x + 4
\][/tex]
5. Factor by grouping:
- Group the terms:
[tex]\[
(3x^2 + 3x) + (4x + 4)
\][/tex]
- Factor out the common factor from each group:
[tex]\[
3x(x + 1) + 4(x + 1)
\][/tex]
- Notice that [tex]\((x + 1)\)[/tex] is a common factor:
[tex]\[
(3x + 4)(x + 1)
\][/tex]
6. Write the complete factorization of the original polynomial:
- Combine the factored form of the GCF and the quadratic expression:
[tex]\[
9x^4 + 21x^3 + 12x^2 = 3x^2(3x + 4)(x + 1)
\][/tex]
Therefore, the polynomial is completely factored as [tex]\(3x^2(3x + 4)(x + 1)\)[/tex].
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients and the variable powers in the terms [tex]\(9x^4\)[/tex], [tex]\(21x^3\)[/tex], and [tex]\(12x^2\)[/tex].
- The GCF of the coefficients [tex]\(9\)[/tex], [tex]\(21\)[/tex], and [tex]\(12\)[/tex] is [tex]\(3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] in all the terms is [tex]\(x^2\)[/tex].
- Therefore, the GCF of the entire polynomial is [tex]\(3x^2\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF, [tex]\(3x^2\)[/tex]:
[tex]\[
9x^4 + 21x^3 + 12x^2 = 3x^2(3x^2 + 7x + 4)
\][/tex]
3. Factor the remaining quadratic expression, [tex]\(3x^2 + 7x + 4\)[/tex]:
- Look for two numbers that multiply to [tex]\(3 \times 4 = 12\)[/tex] and add to [tex]\(7\)[/tex]. These numbers are [tex]\(3\)[/tex] and [tex]\(4\)[/tex].
4. Rewrite the middle term ([tex]\(7x\)[/tex]) using [tex]\(3\)[/tex] and [tex]\(4\)[/tex]:
- Rewrite [tex]\(7x\)[/tex] as [tex]\(3x + 4x\)[/tex]:
[tex]\[
3x^2 + 7x + 4 = 3x^2 + 3x + 4x + 4
\][/tex]
5. Factor by grouping:
- Group the terms:
[tex]\[
(3x^2 + 3x) + (4x + 4)
\][/tex]
- Factor out the common factor from each group:
[tex]\[
3x(x + 1) + 4(x + 1)
\][/tex]
- Notice that [tex]\((x + 1)\)[/tex] is a common factor:
[tex]\[
(3x + 4)(x + 1)
\][/tex]
6. Write the complete factorization of the original polynomial:
- Combine the factored form of the GCF and the quadratic expression:
[tex]\[
9x^4 + 21x^3 + 12x^2 = 3x^2(3x + 4)(x + 1)
\][/tex]
Therefore, the polynomial is completely factored as [tex]\(3x^2(3x + 4)(x + 1)\)[/tex].
