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Answer :
To factor the polynomial [tex]\(24x^{11} + 4x^{10} - 6x^9 + 2x^8\)[/tex] using the greatest common factor (GCF), follow these steps:
1. Identify the GCF of the coefficients:
Look at the numerical coefficients of each term: 24, 4, -6, and 2. The greatest common factor of these numbers is 2.
2. Identify the GCF of the variables:
Look at the variable part of each term, which is [tex]\(x^{11}\)[/tex], [tex]\(x^{10}\)[/tex], [tex]\(x^9\)[/tex], and [tex]\(x^8\)[/tex]. The smallest power of [tex]\(x\)[/tex] among these is [tex]\(x^8\)[/tex].
3. Overall GCF:
Combine the GCF of the coefficients and the variables. The GCF of the entire polynomial is [tex]\(2x^8\)[/tex].
4. Factor out the GCF:
Divide each term of the polynomial by [tex]\(2x^8\)[/tex]:
[tex]\[
\begin{align*}
24x^{11} &\div 2x^8 = 12x^3, \\
4x^{10} &\div 2x^8 = 2x^2, \\
-6x^9 &\div 2x^8 = -3x, \\
2x^8 &\div 2x^8 = 1.
\end{align*}
\][/tex]
5. Write the factored form:
After factoring out the GCF [tex]\(2x^8\)[/tex], the polynomial becomes:
[tex]\[
2x^8(12x^3 + 2x^2 - 3x + 1).
\][/tex]
So, the factored form of [tex]\(24x^{11} + 4x^{10} - 6x^9 + 2x^8\)[/tex] is [tex]\(2x^8(12x^3 + 2x^2 - 3x + 1)\)[/tex].
1. Identify the GCF of the coefficients:
Look at the numerical coefficients of each term: 24, 4, -6, and 2. The greatest common factor of these numbers is 2.
2. Identify the GCF of the variables:
Look at the variable part of each term, which is [tex]\(x^{11}\)[/tex], [tex]\(x^{10}\)[/tex], [tex]\(x^9\)[/tex], and [tex]\(x^8\)[/tex]. The smallest power of [tex]\(x\)[/tex] among these is [tex]\(x^8\)[/tex].
3. Overall GCF:
Combine the GCF of the coefficients and the variables. The GCF of the entire polynomial is [tex]\(2x^8\)[/tex].
4. Factor out the GCF:
Divide each term of the polynomial by [tex]\(2x^8\)[/tex]:
[tex]\[
\begin{align*}
24x^{11} &\div 2x^8 = 12x^3, \\
4x^{10} &\div 2x^8 = 2x^2, \\
-6x^9 &\div 2x^8 = -3x, \\
2x^8 &\div 2x^8 = 1.
\end{align*}
\][/tex]
5. Write the factored form:
After factoring out the GCF [tex]\(2x^8\)[/tex], the polynomial becomes:
[tex]\[
2x^8(12x^3 + 2x^2 - 3x + 1).
\][/tex]
So, the factored form of [tex]\(24x^{11} + 4x^{10} - 6x^9 + 2x^8\)[/tex] is [tex]\(2x^8(12x^3 + 2x^2 - 3x + 1)\)[/tex].
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