Question

How do you expand `(5x+2y)^3`?

Answer 1

`(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:
`color(white)(aaaaaaaaaaaaaaaaaaa)1`
`color(white)(aaaaaaaaaaaaaaaaa)1color(white)(aaa)1`
`color(white)(aaaaaaaaaaaaaaa)1color(white)(aaa)2color(white)(aaa)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:
`(5x+2y)^3=(5x)^3+3(5x)^2(2y)+3(5x)(2y)^2+(2y)^3`

`=125x^3+3*25x^2*2y+3*5x*4y^2+8y^3`

`=125x^3+150x^2y+60xy^2+8y^3`