Q. 22

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

$24 y^{3} - 18 y^{2} - 30 y$

Verified

Answer

The factored form is $\begin{array}{rcl} 6 y (4 y^{2} - 3 y - 5) \end{array}$ .

1Step 1. Given Information

The given polynomial is $24 y^{3} - 18 y^{2} - 30 y$ .

2Step 2. Find the GCF of the terms

$24 y^{3} = 2 \cdot 2 \cdot 2 \cdot 3 \cdot y \cdot y \cdot y 18 y^{2} = 2 \cdot 3 \cdot 3 \cdot y \cdot y 30 y = 2 \cdot 3 \cdot 5 \cdot y$

From the obtained factors, it is observed that the common factors are $2, 3, y$ .
Multiply the common factors.

$2 \cdot 3 \cdot y = 6 y$

3Step 3. Rewrite the given expression

$6 y \cdot 4 y^{2} - 6 y \cdot 3 y - 6 y \cdot 5$

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

$6 y \cdot 4 y^{2} - 6 y \cdot 3 y - 6 y \cdot 5 = 6 y (4 y^{2} - 3 y - 5)$

4Step 4. Check

$\begin{array}{rcl} 6 y (4 y^{2} - 3 y - 5) & = & 6 y \cdot 4 y^{2} - 6 y \cdot 3 y - 6 y \cdot 5 \\ = & 24 y^{3} - 18 y^{2} - 30 y \end{array}$

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