Answer :
To solve the problem of factoring the polynomial [tex]\(80x^{13} - 200x^{11} + 125x^9\)[/tex], follow these steps:
1. Identify the Greatest Common Factor (GCF):
Look for the greatest common factor of the coefficients and the lowest power of [tex]\(x\)[/tex] present in all terms.
- Coefficients: 80, 200, and 125
- Common factor of the coefficients: 5 (since 80, 200, and 125 are all divisible by 5)
- Powers of [tex]\(x\)[/tex]: [tex]\(x^{13}\)[/tex], [tex]\(x^{11}\)[/tex], [tex]\(x^9\)[/tex]
- The lowest power of [tex]\(x\)[/tex] present is [tex]\(x^9\)[/tex].
Therefore, the GCF is [tex]\(5x^9\)[/tex].
2. Factor out the GCF:
Divide each term of the polynomial by the GCF [tex]\(5x^9\)[/tex]:
[tex]\[
\begin{align*}
80x^{13} & \rightarrow \frac{80x^{13}}{5x^9} = 16x^4 \\
-200x^{11} & \rightarrow \frac{-200x^{11}}{5x^9} = -40x^2 \\
125x^{9} & \rightarrow \frac{125x^9}{5x^9} = 25
\end{align*}
\][/tex]
So, the polynomial becomes:
[tex]\[
5x^9(16x^4 - 40x^2 + 25)
\][/tex]
3. Factor the Quadratic Expression:
The expression inside the parentheses [tex]\(16x^4 - 40x^2 + 25\)[/tex] is a quadratic expression in terms of [tex]\(x^2\)[/tex]. Let [tex]\(y = x^2\)[/tex]. Then the expression becomes:
[tex]\[
16y^2 - 40y + 25
\][/tex]
This is a quadratic in [tex]\(y\)[/tex]. Factor the quadratic expression:
[tex]\[
\begin{align*}
16y^2 - 40y + 25 &= (4y - 5)^2
\end{align*}
\][/tex]
Substitute back [tex]\(y = x^2\)[/tex]:
[tex]\[
(4x^2 - 5)^2
\][/tex]
4. Combine the Factors:
The complete factorization of the original polynomial is:
[tex]\[
5x^9(4x^2 - 5)^2
\][/tex]
Let's look at the options you were given:
- (A) [tex]\(5x^9(4x^2 + 5)^2\)[/tex]
- (B) [tex]\(5x^9(4x^2 - 5)^2\)[/tex]
- (C) [tex]\(5x^9(4x^2 - 5)(4x^2 + 5)\)[/tex]
- (D) [tex]\(x^9(40x^2 - 5)^2\)[/tex]
Comparing our factored form, [tex]\(5x^9(4x^2 - 5)^2\)[/tex], to the options, you can see the correct answer is option (B).
1. Identify the Greatest Common Factor (GCF):
Look for the greatest common factor of the coefficients and the lowest power of [tex]\(x\)[/tex] present in all terms.
- Coefficients: 80, 200, and 125
- Common factor of the coefficients: 5 (since 80, 200, and 125 are all divisible by 5)
- Powers of [tex]\(x\)[/tex]: [tex]\(x^{13}\)[/tex], [tex]\(x^{11}\)[/tex], [tex]\(x^9\)[/tex]
- The lowest power of [tex]\(x\)[/tex] present is [tex]\(x^9\)[/tex].
Therefore, the GCF is [tex]\(5x^9\)[/tex].
2. Factor out the GCF:
Divide each term of the polynomial by the GCF [tex]\(5x^9\)[/tex]:
[tex]\[
\begin{align*}
80x^{13} & \rightarrow \frac{80x^{13}}{5x^9} = 16x^4 \\
-200x^{11} & \rightarrow \frac{-200x^{11}}{5x^9} = -40x^2 \\
125x^{9} & \rightarrow \frac{125x^9}{5x^9} = 25
\end{align*}
\][/tex]
So, the polynomial becomes:
[tex]\[
5x^9(16x^4 - 40x^2 + 25)
\][/tex]
3. Factor the Quadratic Expression:
The expression inside the parentheses [tex]\(16x^4 - 40x^2 + 25\)[/tex] is a quadratic expression in terms of [tex]\(x^2\)[/tex]. Let [tex]\(y = x^2\)[/tex]. Then the expression becomes:
[tex]\[
16y^2 - 40y + 25
\][/tex]
This is a quadratic in [tex]\(y\)[/tex]. Factor the quadratic expression:
[tex]\[
\begin{align*}
16y^2 - 40y + 25 &= (4y - 5)^2
\end{align*}
\][/tex]
Substitute back [tex]\(y = x^2\)[/tex]:
[tex]\[
(4x^2 - 5)^2
\][/tex]
4. Combine the Factors:
The complete factorization of the original polynomial is:
[tex]\[
5x^9(4x^2 - 5)^2
\][/tex]
Let's look at the options you were given:
- (A) [tex]\(5x^9(4x^2 + 5)^2\)[/tex]
- (B) [tex]\(5x^9(4x^2 - 5)^2\)[/tex]
- (C) [tex]\(5x^9(4x^2 - 5)(4x^2 + 5)\)[/tex]
- (D) [tex]\(x^9(40x^2 - 5)^2\)[/tex]
Comparing our factored form, [tex]\(5x^9(4x^2 - 5)^2\)[/tex], to the options, you can see the correct answer is option (B).
