randVar() randVar() randRangeExclude(-9, 9, [-1, 0, 1]) -A * A

Simplify the right-hand side of the equation below and state the condition under which the simplification is valid.

`Y = \dfrac{X^2 + CONSTANT}{X + A}`

`Y =`
X - A
`\space X \neq \space` -A

The numerator is in the form `\pink{a^2} - \blue{b^2}`, which is a difference of two squares so we can factor it as `(\pink{a} + \blue{b})(\pink{a} - \blue{b})`.

`\qquad \pink{a = X}`

`\qquad \blue{b = \sqrt{A * A} = abs(A)}`

So we can rewrite the expression as:

`Y = \dfrac{(\pink{X} + \blue{A})(\pink{X} \blue{-A})} \dfrac{(\pink{X} \blue{A})(\pink{X} + \blue{-A})} {X + A}`

We can divide the numerator and denominator by `(X + A)`:

`Y = \dfrac{\cancel{(\pink{X} + \blue{A})}(\pink{X} \blue{-A})} \dfrac{\cancel{(\pink{X} \blue{A})}(\pink{X} + \blue{-A})} {\cancel{X + A}} = X + -A`

Because we divided by `(X + A)`,

`\begin{align} X + A &\neq 0 \\ X &\neq -A \end{align}`

Therefore,

`\qquad Y = X - A`
`\qquad X \neq -A`

randVar() randVar() randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) A * B -A - B

Simplify the right-hand side of the equation below and state the condition under which the simplification is valid.

`Y = \dfrac{X^2 + plus(LINEAR + X) + CONSTANT}{X - A}`

`Y =`
X - B
`\space X \neq \space` A

`X^2 + plus(LINEAR + X) + CONSTANT = (X - A)(X - B)`

So we can rewrite the expression as:

`Y = \dfrac{(X + -A)(X + -B)}{X + -A}`

We can divide the numerator and denominator by `(X - A)`:

`Y = \dfrac{\cancel{(X - A)}(X - B)}{\cancel{X - A}} = X - B`

Because we divided by `(X - A)`,

`\begin{align} X + -A &\neq 0 \\ X &\neq A \end{align}`

Therefore,

`\qquad Y = X - B`
`\qquad X \neq A`

First factor the polynomial in the numerator.