Chapter 4.3, Problem 81HP

To determine

To calculate: The factors of the polynomial $40x^5-135x^2{y}^3$ .

Answer to Problem 81HP

The factor of the polynomial is $5x^2\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)$ .

Explanation of Solution

Given information:

The polynomial $40x^5-135x^2{y}^3$ .

Formula used:

Factorization formula for difference of cubes $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$ .

Calculation:

Consider the given polynomial $40x^5-135x^2{y}^3$ .

Recall that factorization formula for difference of cubes $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$ .

First take out greatest common factor from the polynomial and then apply the rule for factorization for difference of cubes.

So, $40x^5-135x^2{y}^3$ will be factorized as,

  $\begin{array}{c}40x^5-135x^2{y}^3=5x^2\left(8x^3-27y^3\right)\\ =5x^2\left({\left(2x\right)}^3-{\left(3y\right)}^3\right)\\ =5x^2\left(2x-3y\right)\left({\left(2x\right)}^2+\left(2x\right)\left(3y\right)+{\left(3y\right)}^2\right)\\ =5x^2\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)\end{array}$ .

Thus, the factor of the polynomial is $5x^2\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)$ .