randVar() randVar() randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) randRangeExclude(-10, 10, [-1, 0, 1]) A * B -A - B new RationalExpression([[C, X], C * -B])

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

`Y = \dfrac{CX^2 + C * LINEARX + C * CONSTANT}{X - A}`

`Y =`
C X + C * (-B)
C(X + (-B))
`X \neq` A

First factor the polynomial in the numerator.

We notice that all the terms in the numerator have a common factor of `C`, so we can rewrite the expression:

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

Then we factor the remaining polynomial:

`X^2 LINEAR > 0 ? "+" : "" \green{LINEAR}X CONSTANT > 0 ? "+" : "" \blue{CONSTANT}`

`\pink{-A} B < 0 ? "+" : "" \pink{-B} = \green{LINEAR}`

`\pink{-A} \times \pink{-B} = \blue{CONSTANT}`

`(X A < 0 ? "+" : "" \pink{-A}) (X B < 0 ? "+" : "" \pink{-B})`

This gives us a factored expression:

`Y = \dfrac{C(X A < 0 ? "+" : "" \pink{-A}) (X B < 0 ? "+" : "" \pink{-B})}{X + -A}`

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

`Y = \dfrac{C\cancel{(X A < 0 ? "+" : "" \pink{-A})} (X B < 0 ? "+" : "" \pink{-B})}{\cancel{X - A}} = C(X + -B)`

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

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

Therefore,

`\qquad Y = C(X - B)`
`\qquad X \neq A`