Question

Simplify ${\left(2x-5y\right)}^3-{\left(2x+5y\right)}^3$

Answer 1

Step 1: Simplify the term ${\left(2x-5y\right)}^3$.

Algebraic identities.

The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials.

Use the algebraic identity ${\left(a-b\right)}^3=a^3-b^3-3ab\left(a-b\right)$ to simplify the expression ${\left(2x-5y\right)}^3$:

$\begin{array}{rcl}{\left(2x-5y\right)}^3& =& {\left(2x\right)}^3-{\left(5y\right)}^3-3\left(2x\right)\left(5y\right)\left(2x-5y\right)\\ & =& 8x^3-125y^3-30xy\left(2x-5y\right)\\ & =& 8x^3-125y^3-60x^{2}y+150x{y}^2\end{array}$

$\therefore {\left(2x-5y\right)}^3=8x^3-125y^3-60x^{2}y+150x{y}^2$

Step 2: Simplify the term ${\left(2x+5y\right)}^3$.

Use the algebraic identity ${\left(a+b\right)}^3=a^3+b^3+3ab\left(a+b\right)$ to simplify the expression ${\left(2x+5y\right)}^3$:

$\begin{array}{rcl}{\left(2x+5y\right)}^3& =& {\left(2x\right)}^3+{\left(5y\right)}^3+3\left(2x\right)\left(5y\right)\left(2x+5y\right)\\ & =& 8x^3+125y^3+30xy\left(2x+5y\right)\\ & =& 8x^3+125y^3+60x^{2}y+150x{y}^2\end{array}$

$\therefore {\left(2x+5y\right)}^3=\begin{array}{rcl}& & 8\end{array}\begin{array}{rcl}& & x\end{array}\begin{array}{rcl}& & \end{array}$

Step 3: Simplify the given expression ${\left(2x-5y\right)}^3-{\left(2x+5y\right)}^3$:

Use the results obtained in Steps 1 and 2 to simplify the expression ${\left(2x-5y\right)}^3-{\left(2x+5y\right)}^3$:

$\begin{array}{rcl}{\left(2x-5y\right)}^3-{\left(2x+5y\right)}^3& =& \left(8x^3-125y^3-60x^{2}y+150x{y}^2\right)-\left(8x^3+125y^3+60x^{2}y+150x{y}^2\right)\\ & =& 8x^3-125y^3-60x^{2}y+150x{y}^2-8x^3-125y^3-60x^{2}y-150x{y}^2\\ & =& 8x^3-8x^3-125y^3-125y^3-60x^{2}y-60x^{2}y+150x{y}^2-150x{y}^2\\ & =& -250y^3-120x^{2}y\end{array}$

Hence, ${\left(2x-5y\right)}^3-{\left(2x+5y\right)}^3=-250y^3-120x^{2}y$.