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)

Answer 1

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:

\(\boxed{4x(5x + 6)(x - 6)}\)