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

Dec 12, 2015

$f \left(x\right) = \left({x}^{3} - 2 {x}^{2} y + x {y}^{2} - 2 {x}^{2} - 2 {y}^{2} + 2 x y\right)$

Explanation:

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

$f \left(x\right) = \left(x - 2\right) {\left(x - y\right)}^{2}$

$f \left(x\right) = \left(x - 2\right) \left(x - y\right) \left(x - y\right)$

$f \left(x\right) = \left(x - 2\right) \left({x}^{2} - x y - x y + {y}^{2}\right)$

$f \left(x\right) = \left(x - 2\right) \left({x}^{2} - 2 x y + {y}^{2}\right)$

$f \left(x\right) = \left({x}^{3} - 2 {x}^{2} y + x {y}^{2} - 2 {x}^{2} + 4 x y - 2 {y}^{2}\right)$

$f \left(x\right) = \left({x}^{3} - 2 {x}^{2} y + x {y}^{2} - 2 {x}^{2} - 2 {y}^{2} + 4 x y\right)$

Dec 15, 2015

$\textcolor{g r e e n}{{x}^{3} - 2 {x}^{2} - 2 {x}^{2} y + 4 x y + x {y}^{2} - 2 {y}^{2}}$

Attempted to make clear what is happening by using color

Explanation:

Given: $\left(x - 2\right) {\left(x - y\right)}^{2.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(1\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider ${\left(x - y\right)}^{2}$

Write as $\textcolor{b r o w n}{\textcolor{b l u e}{\left(x - y\right)} \left(x - y\right)}$

This is distributive so we have:

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

$\textcolor{b r o w n}{\textcolor{b l u e}{x} \left(x - y\right) \textcolor{b l u e}{- y} \left(x - y\right)}$

Giving:

${x}^{2} - x y - x y + {y}^{2}$

${x}^{2} - 2 x y + {y}^{2.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(2\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute (2) into (1) for ${\left(x - y\right)}^{2}$ 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:

${x}^{3} - 2 {x}^{2} y + x {y}^{2} - 2 {x}^{2} + 4 x y - 2 {y}^{2}$

Changing the order giving x precedence over y

$\textcolor{g r e e n}{{x}^{3} - 2 {x}^{2} - 2 {x}^{2} y + 4 x y + x {y}^{2} - 2 {y}^{2}}$