Question
Simplify the following: $(2x - 5y)^3 - (2x + 5y)^3$
✓
Answer
Given $(2x - 5y)^3 - (2x + 5y)^3 $
We shall use the identity $a^3 - b^3 = (a - b)(a^2 + b^2 + ab)$
Here $a = (2x - 5y), b = (2x + 5y)$ By applying the identity
we get $=\big(2\text{x}-5\text{y}-2\text{x}+5\text{y}\big)$
$\Big[(2\text{x}-5\text{y})^2+(2\text{x}+5\text{y})^2\big((2\text{x}-5\text{y})\times(2\text{x}+5\text{y})\big)\Big]$
$=\big(2\text{x}-5\text{y}-2\text{x}-5\text{y})\Big[2\text{x}\times2\text{x}+5\text{y}\times5\text{y}-2\times2\text{x}\times5\text{y}\big)$
$+\big(2\text{x}\times2\text{x}+5\text{y}\times5\text{y}+2\times2\text{x}\times5\text{y}\big)+\big(4\text{x}^2-25\text{y}^2\big)\Big]$
$=(-10\text{y})\Big[\big(4\text{x}^2+25\text{y}^2-20\text{xy}\big)$
$+\big(4\text{x}^2+25\text{y}^2+20\text{xy}\big)+4\text{x}^2-25\text{y}^2\Big]$
$=(-10\text{y})\Big[4\text{x}^2+25\text{y}^2-20\text{xy}+4\text{x}^2+25\text{y}^2+20\text{xy}+4\text{x}^2-25\text{y}^2\Big]$ By rearranging the variable
we get, $=(-10\text{y})\Big[4\text{x}^2+4\text{x}^2+4\text{x}^2+25\text{y}^2\Big]$
$=-10\text{y}\times\big[12\text{x}^2+25\text{y}^2\big]$
$=-120\text{x}^2\text{y}-250\text{y}^3$
Hence the value of $\big(2\text{x}-5\text{y}\big)^3-\big(2\text{x}+5\text{y}\big)^3$ is
$-120\text{x}^2\text{y}-250\text{y}^3.$
We shall use the identity $a^3 - b^3 = (a - b)(a^2 + b^2 + ab)$
Here $a = (2x - 5y), b = (2x + 5y)$ By applying the identity
we get $=\big(2\text{x}-5\text{y}-2\text{x}+5\text{y}\big)$
$\Big[(2\text{x}-5\text{y})^2+(2\text{x}+5\text{y})^2\big((2\text{x}-5\text{y})\times(2\text{x}+5\text{y})\big)\Big]$
$=\big(2\text{x}-5\text{y}-2\text{x}-5\text{y})\Big[2\text{x}\times2\text{x}+5\text{y}\times5\text{y}-2\times2\text{x}\times5\text{y}\big)$
$+\big(2\text{x}\times2\text{x}+5\text{y}\times5\text{y}+2\times2\text{x}\times5\text{y}\big)+\big(4\text{x}^2-25\text{y}^2\big)\Big]$
$=(-10\text{y})\Big[\big(4\text{x}^2+25\text{y}^2-20\text{xy}\big)$
$+\big(4\text{x}^2+25\text{y}^2+20\text{xy}\big)+4\text{x}^2-25\text{y}^2\Big]$
$=(-10\text{y})\Big[4\text{x}^2+25\text{y}^2-20\text{xy}+4\text{x}^2+25\text{y}^2+20\text{xy}+4\text{x}^2-25\text{y}^2\Big]$ By rearranging the variable
we get, $=(-10\text{y})\Big[4\text{x}^2+4\text{x}^2+4\text{x}^2+25\text{y}^2\Big]$
$=-10\text{y}\times\big[12\text{x}^2+25\text{y}^2\big]$
$=-120\text{x}^2\text{y}-250\text{y}^3$
Hence the value of $\big(2\text{x}-5\text{y}\big)^3-\big(2\text{x}+5\text{y}\big)^3$ is
$-120\text{x}^2\text{y}-250\text{y}^3.$
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