Chapter 6.5, Problem 52PPS

To determine

To find: the factor of the given polynomial and check whether the polynomial is factored or prime.

Answer to Problem 52PPS

  $8x^3-27y^3=x^2{y}^2\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)$ .

Explanation of Solution

Given:

The given polynomial is $8x^5{y}^2-27x^2{y}^5$ .

Calculation:

Now, to find the factor of the given polynomial. Therefore,

  $\begin{array}{l}8x^5{y}^2-27x^2{y}^5\left\{\text{given polynomial}\right\}\\ =x^2{y}^2\left(8x^3-27y^3\right)\left\{\text{factor out the GCF}\right\}\end{array}$

Since are both perfect squares, so write the factor of the differences of two square.

  $\begin{array}{l}8x^3-27y^3={\left(2x\right)}^3-{\left(3y\right)}^3\\ a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\\ =\left(2x-3y\right)\left({\left(2x\right)}^2+\left(2x\right)\left(3y\right)+{\left(3y\right)}^2\right)\left\{\text{simplified form}\right\}\\ =\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)\end{array}$

Thus, $8x^3-27y^3=x^2{y}^2\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)$

Hence, the factor of the given polynomial $8x^3-27y^3=x^2{y}^2\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)$ .