Rational Expressions are fractions comprised of polynomials. Basic artihmetic operations such as addition, subtraction, multiplication, and division can be performed on these expressions. Doing so will always result in another rational expression.
Restrictions are values that aren't part of the expressions domain. Substituting these values into any variable of the denominator will set it to \(0\); this will consequently make the expression undefined. It's important to state these values explicitly when evaluating or graphing the expression.
The process for simplifying rational expressions works as such:
NOTE: Do not cancel parts of the polynomial (ie one term of a binomial). Otherwise, the expression will not be properly simplified!!
Simplify the expression \(\cfrac{3x}{3x²-6x}\) and state its restrictions.
First, we can factor the denominator as the numerator cannot be factored any further:
We can now check for restrictions. To do so, we must identify any values that would cause the denominator to be set to \(0\):
\(x \neq 0\)
\(x - 2 \neq 0\)
\(x \neq 2\)
Finally, we can cancel identical polynomials to get our simplified expression:
\(= \cfrac{\cancel{3x}}{\cancel{3x}(x-2)}\)
\(= \cfrac{1}{(x-2)}\)
Therefore, we can determine that the simplified expression is \(\boldsymbol{\cfrac{1}{(x-2)}}\) and the restrictions are \(\boldsymbol{x \neq 0, 2}\).
\(\cfrac{2x²-13x+6}{x²-36}\)
First, we can factor the numerator and denominator. We can factor the numerator through sum product factoring and the denominator by finding the difference of squares:
\(= \cfrac{2x²\textcolor{blue}{-13x}+6}{\textcolor{green}{x²-36}}\)
\(= \cfrac{2x²\textcolor{blue}{-12x-x}+6}{\textcolor{green}{(x+6)(x-6)}}\)
\(= \cfrac{\textcolor{purple}{2x}\textcolor{red}{(x-6)}-\textcolor{purple}{1}\textcolor{red}{(x-6)}}{(x+6)(x-6)}\)
\(= \cfrac{\textcolor{purple}{(2x-1)}\textcolor{red}{(x-6)}}{(x+6)(x-6)}\)
We can now check for restrictions. To do so, we must identify any values that would cause the denominator to be set to \(0\):
\(x + 6 \neq 0\)
\(x \neq -6\)
\(x - 6 \neq 0\)
\(x \neq 6\)
Finally, we can cancel identical polynomials to get our simplified expression:
\(= \cfrac{(2x-1)(\cancel{x-6})}{(x+6)(\cancel{x-6})}\)
\(= \cfrac{2x-1}{x+6}\)
Therefore, we can determine that the simplified expression is \(\boldsymbol{\cfrac{2x-1}{x+6}}\)and the restrictions are \(\boldsymbol{x ≠ \pm 6}\).
\(\cfrac{42x³y⁵}{-12x⁴y²}\)
Since the numerator and denominator can't be fully factored, we can check for restrictions:
\(x \neq 0\)
\(y \neq 0\)
Finally, we can cancel identical polynomials to get our simplified expression:
\(= \cfrac{42x³y⁵}{-12x⁴y²}\)
\(= \cfrac{42\cancel{x³}\cdot \cancel{y²} \cdot y³}{-12\cancel{x³} \cdot x \cdot \cancel{y²}}\)
\(= \cfrac{-7y³}{2x}\)
Therefore, we can determine that the simplified expression is \(\boldsymbol{\cfrac{-7y^3}{2x}}\) and the restrictions are \(\boldsymbol{x ≠ 0}\) and \(\boldsymbol{y ≠ 0}\).
First, we can factor the numerator and denominator. We can factor the both through decomposition:
\(= \cfrac{2x^2\textcolor{blue}{+xy}-3y^2}{2y^2\textcolor{red}{-xy}-x^2}\)
\(= \cfrac{2x^2\textcolor{blue}{+3xy-2xy}-3y^2}{2y^2\textcolor{red}{-2xy+xy}-x^2}\)
\(= \cfrac{2x\textcolor{green}{(x-y)}+3y\textcolor{green}{(x-y)}}{2y\textcolor{magenta}{(y-x)}+x\textcolor{magenta}{(y-x)}}\)
\(= \cfrac{(2x+3y)\textcolor{green}{(x-y)}}{(2y+x)\textcolor{magenta}{(y-x)}}\)
\(= \cfrac{(2x+3y)(x-y)}{-(x-y)(2y+x)}\)
We can now check for restrictions. To do so, we must identify any values that would cause the denominator to be set to \(0\):
We can first evaluate the restriction for \(x-y\):
\(x - y ≠ 0\)
\(x \neq y \; \text{OR} \; y \neq x\)
We can then evaluate the restriction for \(2y+x\):
\(2y+x ≠ 0\)
\(\cfrac{\cancel{2}y}{\cancel2} ≠ \cfrac{-x}{2}\)
\(y ≠ \cfrac{-x}{2} \; \text{OR} \; x ≠ -2y\)
Finally, we can cancel identical polynomials to get our simplified expression:
Therefore, we can determine that the simplified expression is \(\boldsymbol{\cfrac{-(2x+3y)}{2y+x}}\).
Additionally, the restrictions can be divided into two sets: