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 Dividing polynomials by binomials 2
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 =
CX + 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)