Questions: Pearson MyLab and Mas Do Homework - Section mylab.pearson.com Question 16, 6.3.55 Factor completely. -6 x^3+3 x^2+45 x Select the correct choice below and fill in any answer boxes within your choice. A. -6 x^3+3 x^2+45 x= B. The polynomial is prime.

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Question 16, 6.3.55

Factor completely.
-6 x^3+3 x^2+45 x

Select the correct choice below and fill in any answer boxes within your choice.
A. -6 x^3+3 x^2+45 x=
B. The polynomial is prime.

Transcript text: Pearson MyLab and Mas Do Homework - Section mylab.pearson.com Question 16, 6.3.55 Factor completely. \[ -6 x^{3}+3 x^{2}+45 x \] Select the correct choice below and fill in any answer boxes within your choice. A. $-6 x^{3}+3 x^{2}+45 x=$ $\square$ B. The polynomial is prime.

Solution

Solution Steps

To factor the given polynomial completely, we first look for the greatest common factor (GCF) of all the terms. Then, we factor out the GCF and check if the remaining polynomial can be factored further.

Solution Approach

Identify the GCF of the terms in the polynomial.
Factor out the GCF.
Check if the remaining polynomial can be factored further.

Step 1: Identify the Polynomial

We start with the polynomial: \[ -6x^3 + 3x^2 + 45x \]

Step 2: Factor Out the Greatest Common Factor (GCF)

The GCF of the terms $-6x^3$, $3x^2$, and $45x$ is $-3x$. We factor this out: \[ -6x^3 + 3x^2 + 45x = -3x(2x^2 - x - 15) \]

Step 3: Factor the Quadratic Expression

Next, we need to factor the quadratic expression $2x^2 - x - 15$. We can find two numbers that multiply to $2 \cdot (-15) = -30$ and add to $-1$. The numbers $-6$ and $5$ work: \[ 2x^2 - 6x + 5x - 15 = 2x(x - 3) + 5(x - 3) = (2x + 5)(x - 3) \]

Step 4: Combine the Factors

Now we can write the complete factorization of the original polynomial: \[ -6x^3 + 3x^2 + 45x = -3x(2x + 5)(x - 3) \]