Questions: Question 16 Factor the following expression completely: 48 x^3 + 4 x^2 - 24 x Hint: Factor out the GCF first!

Transcript text: Question 16 Factor the following expression completely: \[ 48 x^{3}+4 x^{2}-24 x= \] $\square$ Hint: Factor out the GCF first! Question Help: Message instructor Submit Question Jump to Answer

Solution

Solution Steps

To factor the given expression completely, first identify the greatest common factor (GCF) of all the terms. Then, factor out the GCF from the expression. After factoring out the GCF, look at the remaining expression and determine if it can be factored further, such as by recognizing it as a quadratic or using other factoring techniques.

Step 1: Identify the Expression

We start with the expression: \[ 48x^3 + 4x^2 - 24x \]

Step 2: Factor Out the GCF

The greatest common factor (GCF) of the terms $48x^3$, $4x^2$, and $-24x$ is $4x$. We factor this out: \[ 48x^3 + 4x^2 - 24x = 4x(12x^2 + x - 6) \]

Step 3: Factor the Quadratic Expression

Next, we need to factor the quadratic expression $12x^2 + x - 6$. We can find two numbers that multiply to $12 \times -6 = -72$ and add to $1$. The numbers $9$ and $-8$ work: \[ 12x^2 + 9x - 8x - 6 = (12x^2 + 9x) + (-8x - 6) \] Factoring by grouping gives: \[ 3x(4x + 3) - 2(4x + 3) = (3x - 2)(4x + 3) \]

Step 4: Combine All Factors

Now we can combine all the factors: \[ 48x^3 + 4x^2 - 24x = 4x(3x - 2)(4x + 3) \]