Questions: Factor the following polynomial completely. 9 x^3 + 63 x^2 + 54 x A. 9 x(x-1)(x-6) B. 9 x(x-9)(x+6) C. 9 x(x+1)(x+6) D. 9 x(x+9)(x+6)

Transcript text: Factor the following polynomial completely. \[ 9 x^{3}+63 x^{2}+54 x \] A. $9 x(x-1)(x-6)$ B. $9 x(x-9)(x+6)$ C. $9 x(x+1)(x+6)$ D. $9 x(x+9)(x+6)$

Solution

Solution Steps

To factor the polynomial $9x^3 + 63x^2 + 54x$ completely, we first look for the greatest common factor (GCF) of all the terms. Then, we factor out the GCF and factor the remaining polynomial if possible.

Step 1: Identify the Polynomial

We start with the polynomial $9x^3 + 63x^2 + 54x$.

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

The GCF of the terms $9x^3$, $63x^2$, and $54x$ is $9x$. We factor this out: \[ 9x^3 + 63x^2 + 54x = 9x(x^2 + 7x + 6) \]

Step 3: Factor the Quadratic Expression

Next, we need to factor the quadratic expression $x^2 + 7x + 6$. We look for two numbers that multiply to $6$ and add to $7$. These numbers are $1$ and $6$. Thus, we can factor the quadratic as: \[ x^2 + 7x + 6 = (x + 1)(x + 6) \]

Step 4: Combine the Factors

Now, we can combine the factors: \[ 9x(x^2 + 7x + 6) = 9x(x + 1)(x + 6) \]