Chapter SH, Problem 5.13EP

To determine

Find the complete factored form of the given polynomial. If it is not then write prime.

Answer to Problem 5.13EP

The factored form of the given polynomial is $\left(3a-5d\right)\left(9a^2+15ad+25d^2\right)$ .

Explanation of Solution

Given:

The given polynomial is $27a^3-125d^3$ .

Calculation:

  $27a^3-125d^3$

The both terms of the polynomial are a perfect cube, so find the factor of the difference of two cubes.

  $\begin{array}{l}{\left(3a\right)}^3-{\left(5d\right)}^3\\ =\left(3a-5d\right)\left[{\left(3a\right)}^2+\left(3a\right)\left(5d\right)+{\left(5d\right)}^2\right]\\ =\left(3a-5d\right)\left[9a^2+15ad+25d^2\right]\\ =\left(3a-5d\right)\left(9a^2+15ad+25d^2\right)\end{array}$

Hence the factored form of the given polynomial is $\left(3a-5d\right)\left(9a^2+15ad+25d^2\right)$ .