Question

What is the standard form of `y= (x-1)^3-(x-3)^2`?

Answer 1

`y=x^3-4x^2+9x-10`

Explanation:

Calculate each part separately:

`(x-1)^3=overbrace((x-1)(x-1))^"multiply just these first"(x-1)`
`=(x^2-x-x+1)(x-1)`
`=(x^2-2x+1)(x-1)`
`=x^3-x^2-2x^2+2x+x-1`
`=x^3-3x^2+3x-1`

`(x-3)^2=(x-3)(x-3)`
`=x^2-3x-3x+9`
`=x^2-6x+9`

Put back into the original expression.

`y=(x^3-3x^2+3x-1)-(x^2-6x+9)`
`=color(red)(x^3)color(blue)(-3x^2)color(green)(+3x)color(purple)(-1)color(blue)(-x^2)color(green)(+6x)color(purple)(-9)`
`=x^3-4x^2+9x-10`