Question

Use the Distributive Property to factor each polynomial.

$8xy-16x^{3}y+10y$

Short answer

The factorization of the given polynomial is$\mathbf{2}\mathit{y}\left(4x-8x^3+5\right)$

Step-by-step solution

Step 1. Find the greatest common factor of 8xy,16x3y and 10y.

Find the factorization of $8xy,16x^{3}y$and $10y$.

$$\begin{gathered} 8xy=2\cdot 2\cdot 2\cdot x\cdot y \\ 16x^{3}y=2\cdot 2\cdot 2\cdot 2\cdot x\cdot x\cdot x\cdot y \\ 10y=2\cdot 5\cdot y \end{gathered}$$

From the factorization of $8xy,16x^{3}y$and $10y$, it can be noticed that the greatest common factor of $8xy,16x^{3}y$and $10y$is$2\cdot y$ or $2y$.

Therefore, the greatest common factor of $8xy,16x^{3}y$and $10y$is $2y$.

Step 2. Write each term as the product of the greatest common factor and the remaining factors.

Therefore, it is obtained that:

$8xy-16x^{3}y+10y=2y\left(4x\right)-2y\left(8x^3\right)+2y\left(5\right)$

Step 3. Use the distributive property to factor out the greatest common factor.

The distributive property states that:

$ab+ac=a\left(b+c\right)$

Now, use the distributive property to factor out the greatest common factor.

Therefore, it is obtained that:

$$\begin{gathered} 8xy-16x^{3}y+10y=2y\left(4x\right)-2y\left(8x^3\right)+2y\left(5\right) \\ =2y\left(4x-8x^3+5\right) \end{gathered}$$

Therefore, the factorization of the given polynomial is $2y\left(4x-8x^3+5\right)$.