Q. 212 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$(y-5{)}^3+125y^3$

Step-by-Step Solution

Answer

The factored form of the given expression is $(6y-5)(21y^2+15y+25)$.

Step 1. Given information.

The given expression is:
$(y-5{)}^3+125y^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}(y-5{)}^3+125y^3& =& (y-5{)}^3+{\left(5y\right)}^3 \\ & =& (y-5+5y)\left({{(y-5)}^2-(y-5)\left(5y\right)+\left(5y\right)}^2\right) \left[\because a^3+b^3=(a+b)(a^2-ab+b^2)\right]\\ & =& (6y-5)(y^2-10y+25-5y^2+25y+25y^2) \left[\because {(a-b)}^2=a^2-2ab+b^2\right]\\ & =& (6y-5)(21y^2+15y+25)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(6y-5)(21y^2+15y+25)$.