Question

Determine the linear factors of the third-degree polynomial 20x3−96x2−144x .(1 point)

Answer 1

To factor the polynomial \( 20x^3 - 96x^2 - 144x \), let's first factor out the greatest common factor (GCF).

The GCF of the coefficients \( 20, -96, \) and \( -144 \) is \( 4 \). Additionally, all terms have an \( x \) factor. The GCF is therefore \( 4x \).

We can factor \( 4x \) out of the polynomial:

\[
20x^3 - 96x^2 - 144x = 4x(5x^2 - 24x - 36)
\]

Next, we need to factor the quadratic \( 5x^2 - 24x - 36 \). To do this, we can use the AC method or look for two numbers that multiply to \( 5 \cdot (-36) = -180 \) and add up to \( -24 \).

The two numbers that fit this requirement are \( -30 \) and \( 6 \), since:

\[
-30 + 6 = -24 \quad \text{and} \quad -30 \cdot 6 = -180
\]

Now we can rewrite the quadratic:

\[
5x^2 - 30x + 6x - 36
\]

Next, we group the terms:

\[
(5x^2 - 30x) + (6x - 36)
\]

Factoring out the common factors in each group:

\[
5x(x - 6) + 6(x - 6)
\]

Now we can factor out \( (x - 6) \):

\[
(5x + 6)(x - 6)
\]

Putting it all together, we have:

\[
20x^3 - 96x^2 - 144x = 4x(5x + 6)(x - 6)
\]

Thus, the linear factors of the polynomial \( 20x^3 - 96x^2 - 144x \) are:

\[
4x, \quad 5x + 6, \quad \text{and} \quad x - 6
\]