Question

What is the standard form of # f=(x - 2)(x - y)^2 #?

Answer 1

`f(x)=(x^3-2x^2y+xy^2-2x^2-2y^2+2xy)`

Explanation:

To rewrite a function in standard form, expand the brackets:

`f(x)=(x-2)(x-y)^2`

`f(x)=(x-2)(x-y)(x-y)`

`f(x)=(x-2)(x^2-xy-xy+y^2)`

`f(x)=(x-2)(x^2-2xy+y^2)`

`f(x)=(x^3-2x^2y+xy^2-2x^2+4xy-2y^2)`

`f(x)=(x^3-2x^2y+xy^2-2x^2-2y^2+4xy)`

Answer 2

`color(green)(x^3 -2x^2-2x^2y+4xy+xy^2-2y^2)`

Attempted to make clear what is happening by using color

Explanation:

Given: `(x-2)(x-y)^2..........................(1)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider `(x-y)^2`

Write as `color(brown)(color(blue)((x-y))(x-y))`

This is distributive so we have:

Every part of the blue bracket is multiplied by all of the brown bracket:

`color(brown)(color(blue)(x)(x-y)color(blue)(-y)(x-y))`

Giving:

`x^2-xy -xy+y^2`

`x^2-2xy+y^2................................(2)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute (2) into (1) for `(x-y)^2` giving:

`color(brown)(color(blue)((x-2))(x^2-2xy+y^2)`

Every part of the blue bracket is multiplied by all of the brown bracket:

`color(brown)(color(blue)(x)(x^2-2xy+y^2)color(blue)(-2)(x^2-2xy+y^2)`

Giving:

`x^3-2x^2y +xy^2-2x^2+4xy-2y^2`

Changing the order giving x precedence over y

`color(green)(x^3 -2x^2-2x^2y+4xy+xy^2-2y^2)`