Question
Determine the linear factors of the third-degree polynomial 20x3−96x2−144x .(1 point) Responses 4x(5x2−24x−36) 4 x left parenthesis 5 x squared minus 24 x minus 36 right parenthesis 4x(5x+6)(x−6) 4 x left parenthesis 5 x plus 6 right parenthesis left parenthesis x minus 6 right parenthesis 4x(5x−6)(x+6) 4 x left parenthesis 5 x minus 6 right parenthesis left parenthesis x plus 6 right parenthesis (5x+6)(x−6)
Answers
Answered by
GPT-4o mini
To factor the polynomial \( 20x^3 - 96x^2 - 144x \), we start by factoring out the greatest common factor:
1. First, identify the common factors in all terms. The greatest common factor of \( 20x^3 \), \( -96x^2 \), and \( -144x \) is \( 4x \).
2. Factor out \( 4x \):
\[
20x^3 - 96x^2 - 144x = 4x(5x^2 - 24x - 36).
\]
Now we need to factor the quadratic \( 5x^2 - 24x - 36 \):
3. To factor \( 5x^2 - 24x - 36 \), we look for factors of \( 5 \cdot (-36) = -180 \) that add up to \( -24 \). The pair of factors that work is \( -30 \) and \( 6 \).
4. Rewrite the quadratic using these factors:
\[
5x^2 - 30x + 6x - 36.
\]
5. Now, factor by grouping:
\[
= (5x^2 - 30x) + (6x - 36)
\]
\[
= 5x(x - 6) + 6(x - 6)
\]
\[
= (5x + 6)(x - 6).
\]
Putting it all together, we have:
\[
20x^3 - 96x^2 - 144x = 4x(5x + 6)(x - 6).
\]
Therefore, the linear factors of the polynomial are:
\(\boxed{4x(5x + 6)(x - 6)}\)
1. First, identify the common factors in all terms. The greatest common factor of \( 20x^3 \), \( -96x^2 \), and \( -144x \) is \( 4x \).
2. Factor out \( 4x \):
\[
20x^3 - 96x^2 - 144x = 4x(5x^2 - 24x - 36).
\]
Now we need to factor the quadratic \( 5x^2 - 24x - 36 \):
3. To factor \( 5x^2 - 24x - 36 \), we look for factors of \( 5 \cdot (-36) = -180 \) that add up to \( -24 \). The pair of factors that work is \( -30 \) and \( 6 \).
4. Rewrite the quadratic using these factors:
\[
5x^2 - 30x + 6x - 36.
\]
5. Now, factor by grouping:
\[
= (5x^2 - 30x) + (6x - 36)
\]
\[
= 5x(x - 6) + 6(x - 6)
\]
\[
= (5x + 6)(x - 6).
\]
Putting it all together, we have:
\[
20x^3 - 96x^2 - 144x = 4x(5x + 6)(x - 6).
\]
Therefore, the linear factors of the polynomial are:
\(\boxed{4x(5x + 6)(x - 6)}\)