Chapter 4.5, Problem 20WE

To determine

To factor thepolynomial.

Answer to Problem 20WE

The value of polynomial is $3x^{2}y\left(y-3\right)\left(y^2+3y+9^2\right)$ .

Explanation of Solution

Given information:

  $3x^2{y}^4-81x^{2}y$

Calculation:

The polynomial can be obtained as

  $\begin{array}{l}3x^2{y}^4-81x^{2}y\\ =3x^2\text{y}\left(y^3-27\right)\left[\text{first take out the common factors}\right]\\ \text{=3}{x}^{2}y\left[{\left(y\right)}^3-{\left(3\right)}^3\right]\left[\text{use sum and differnce of cubes}\right]\\ =3x^{2}y\left(y-3\right)\left(y^2+3y+9^2\right)\left[\text{since},\left(a^3+b^3\right)=\left(a+b\right)\left(a^2-ab+b^2\right)\right]\end{array}$

Hence,

The value of polynomial is $3x^{2}y\left(y-3\right)\left(y^2+3y+9^2\right)$ .