Q. 21

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

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

Verified

Answer

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

1Step 1. Given Information

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

2Step 2. Find the GCF of the terms

$24 x^{3} = 2 \cdot 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot x 12 x^{2} = 2 \cdot 2 \cdot 3 \cdot x \cdot x 15 x = 3 \cdot 5 \cdot x$

$3 \cdot x = 3 x$

3Step 3. Rewrite the given expression

$3 x \cdot 8 x^{2} - 3 x \cdot 4 x + 3 x \cdot 5$

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

$3 x \cdot 8 x^{2} - 3 x \cdot 4 x + 3 x \cdot 5 = 3 x (8 x^{2} - 4 x + 5)$

4Step 4. Check

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

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