Maharashtra BoardEnglish MediumSTD 9MathsAlgebraic Identities4 Marks
Question
Simplify the following:
$(2 x-5 y)^3-(2 x+5 y)^3$
✓
Answer
Given $(2 x-5 y)^3-(2 x+5 y)^3$ We shall use the identity $a^3-b^3=(a-b)\left(a^2+b^2+a b\right)$ Here $a=(2 x-5 y), b=(2 x+5 y)$ By applying the identity we get $=(2 x -5 y -2 x +5 y )$
$\quad\left[(2 x-5 y)^2+(2 x+5 y)^2((2 x-5 y) \times(2 x+5 y))\right]$
$=(2 x-5 y-2 x-5 y)[2 x \times 2 x+5 y \times 5 y-2 \times 2 x \times 5 y)$
$\left.+(2 x \times 2 x+5 y \times 5 y+2 \times 2 x \times 5 y)+\left(4 x^2-25 y^2\right)\right]$
$=(-10 y)\left[\left(4 x^2+25 y^2-20 xy\right)\right.$
$\left.+\left(4 x^2+25 y^2+20 xy\right)+4 x^2-25 y^2\right]$
$=(-10 y)\left[4 x^2+25 y^2-20 xy+4 x^2+25 y^2+20 xy+4 x^2-25 y^2\right]$
By rearranging the variable we get, $=(-10 y )\left[4 x ^2+4 x ^2+4 x ^2+25 y ^2\right]$
$=-10 y \times\left[12 x^2+25 y^2\right]$
$=-120 x^2 y-250 y^3$
Hence the value of $(2 x -5 y )^3-(2 x +5 y )^3$ is $-120 x ^2 y -250 y ^3$.
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