How do you expand ${(5 x + 2 y)}^{3}$ $(5x+2y)^3$ ?

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How do you expand ${(5 x + 2 y)}^{3}$ $(5x+2y)^3$ ?

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Answer 1 · 2017-12-21T20:27:50

${(5 x + 2 y)}^{3} = 125 x^{3} + 150 x^{2} y + 60 x y^{2} + 8 y^{3}$ $(5x+2y)^3=125x^3+150x^2y+60xy^2+8y^3$

Explanation:

We can look at the fourth row (we're counting from zero) of Pascal's triangel to obtain the coefficients of the expansion:
$a a a a a a a a a a a a a a a a a a a 1$ $color(white)(aaaaaaaaaaaaaaaaaaa)1$
$a a a a a a a a a a a a a a a a a 1 a a a 1$ $color(white)(aaaaaaaaaaaaaaaaa)1color(white)(aaa)1$
$a a a a a a a a a a a a a a a 1 a a a 2 a a a 1$ $color(white)(aaaaaaaaaaaaaaa)1color(white)(aaa)2color(white)(aaa)1$
$a a a a a a a a a a a a a 1 a a a 3 a a a 3 a a a 1$ $color(white)(aaaaaaaaaaaaa)1color(white)(aaa)3color(white)(aaa)3color(white)(aaa)1$

Knowing that the exponents of the left term decrease and the exponents of the right increase, we get:
${(5 x + 2 y)}^{3} = {(5 x)}^{3} + 3 {(5 x)}^{2} (2 y) + 3 (5 x) {(2 y)}^{2} + {(2 y)}^{3}$ $(5x+2y)^3=(5x)^3+3(5x)^2(2y)+3(5x)(2y)^2+(2y)^3$

$= 125 x^{3} + 3 \cdot 25 x^{2} \cdot 2 y + 3 \cdot 5 x \cdot 4 y^{2} + 8 y^{3}$ $=125x^3+3*25x^2*2y+3*5x*4y^2+8y^3$

$= 125 x^{3} + 150 x^{2} y + 60 x y^{2} + 8 y^{3}$ $=125x^3+150x^2y+60xy^2+8y^3$

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How do you expand ${(5 x + 2 y)}^{3}$ $(5x+2y)^3$ ?

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