Question

In the following exercises, factor the greatest common factor from each polynomial.

$24x^3-12x^2+15x$

Short answer

The factored form is$\begin{array}{rcl}& & 3x(8x^2-4x+5)\end{array}$.

Step-by-step solution

Step 1. Given Information

The given polynomial is$24x^3-12x^2+15x$.

Step 2. Find the GCF of the terms  

• Factor each of the terms in the given expression.

$$\begin{gathered} 24x^3=2·2·2·3·x·x·x \\ 12x^2=2·2·3·x·x \\ 15x=3·5·x \end{gathered}$$

• From the obtained factors, it is observed that the common factors are $3,x$.
• Multiply the common factors.

$3·x=3x$

• So, the GCF of the terms of the given expression is $3x$.

Step 3. Rewrite the given expression      

• Express each of the term of the given expression as a product of GCF, $3x$.

$3x·8x^2-3x·4x+3x·5$

• Use the reverse distributive property, $ab+ac=a(b+c)$to rewrite the obtained expression.

role="math" localid="1644582020833" $3x·8x^2-3x·4x+3x·5=3x(8x^2-4x+5)$

Step 4. Check

• Multiply the obtained factors.

$\begin{array}{rcl}3x(8x^2-4x+5)& =& 3x·8x^2-3x·4x+3x·5\\ & =& 24x^3-12x^2+15x\end{array}$

• Since the obtained expression is the same as the given expression, the obtained factors are verified.