Q. 211 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$(y-5{)}^3-64y^3$

Step-by-Step Solution

Answer

The factored form of the given expression is $-(3y+5)(21y^2-30y+25)$.

Step 1. Given information.

The given expression is:

$(y-5{)}^3-64y^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}(y-5{)}^3-64y^3& =& (y-5{)}^3-{\left(4y\right)}^3 \\ & =& (y-5-4y)({(y-5)}^2+(y-5\left)\right(4y)+{\left(4y\right)}^2) \left[\because a^3-b^3=(a-b)(a^2+ab+b^2)\right]\\ & =& (-3y-5)(y^2-10y+25+4y^2-20y+16y^2) \left[\because {(a-b)}^2=a^2-2ab+b^2\right]\\ & =& -(3y+5)(21y^2-30y+25)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $-(3y+5)(21y^2-30y+25)$.