9.1: Simplify Rational Expressions
A rational expression is written as a quotient of polynomials
\[\dfrac{P(x)}{Q(x)} \nonumber \]
where \(P(x)\) and \(Q(x)\) are polynomials in one variable \(x\).
To simplify a rational expression, factor both the numerator and the denominator, and remove common factors from both the numerator and the denominator. A simplified rational expression has only one division, and a single numerator and denominator. If the expressions cannot be factored, then the rational expression cannot be simplified.
Simplify the rational expressions:
- \(\dfrac{x^2 + 2x − 3}{x^2 + 4x + 3}\)
- \(\dfrac{(x^2 + 1)^2 (−2) + (2x)(2)(x^2 + 1)(2x)}{(x^2 + 1)^4}\)
- \(\dfrac{(x^2 + 1) \frac{1}{2} (x^{−\frac{1}{2}}) − (2x)(x^{\frac{1}{2}})}{(x^2 + 1)^2}\)
Solution
- \(\begin{array} &&\dfrac{x^2 + 2x − 3}{x^2 + 4x + 3} &\text{Example problem} \\ &\dfrac{(x + 3)(x − 1)}{(x + 3)(x + 1)} &\text{Factor both numerator and denominator.} \\ &\dfrac{\cancel{(x + 3)}(x − 1)}{\cancel{(x + 3)}(x + 1)} &\text{Remove common factors, because \(\dfrac{x + 3}{x + 3} = 1\)} \\ &\dfrac{x − 1}{x + 1} &\text{Final answer} \end{array}\)
- \(\begin{array} &&\dfrac{(x^2 + 1)^2 (−2) + (2x)(2)(x^2 + 1)(2x)}{(x^2 + 1)^4} &\text{Example problem} \\ &\dfrac{2(x^2 + 1)[(x^2 + 1)(−1) + (2x)(2x)]}{(x^2 + 1)^4} &\text{Factor out 2(x^2 + 1)} \\ &\dfrac{2 \cancel{(x^2 + 1)}[(x^2 + 1)(−1) + (2x)(2x)]}{\cancel{(x^2+1)}(x^2 + 1)^3} &\text{Remove common factors, because \(\dfrac{x^2 + 1}{x^2 + 1} = 1\)} \\ &\dfrac{2[−x^2 − 1 + 4x^2]}{(x^2 + 1)^3} &\text{Simplify by multiplying and combining like terms} \\ &\dfrac{2(3x^2 − 1)}{(x^2 + 1)^3} &\text{Final answer} \end{array}\)
- \(\begin{array} &&\dfrac{(x^2 + 1) \frac{1}{2} (x^{−\frac{1}{2}}) − (2x)(x^{\frac{1}{2}})}{(x^2 + 1)^2} &\text{Example problem} \\ &\dfrac{\frac{(x^2+1)}{2x^{\frac{1}{2}}} − (2x)(x^{ \frac{1}{2} })}{(x^2 + 1)^2} &\text{Work with the negative exponent in the first term of the numerator by moving the factor to the denominator of the first term, next to the \(2\).} \\ &\dfrac{(x^2 + 1) − (2x)(x^{\frac{1}{2}} )2(x^{\frac{1}{2}} )}{\dfrac{2x^{\frac{1}{2}} }{(x^2 + 1)^2}} &\text{Common denominator} \\ &\dfrac{x^2 + 1 − 4x^2}{(2x^{\frac{1}{2}} )(x^2 + 1)^2} &\text{Simplify by multiplying and combining like terms} \\ &\dfrac{−3x^2 + 1}{(2x^{\frac{1}{2}} )(x^2 + 1)^2} &\text{Final answer} \end{array}\)
Simplify the rational expressions:
- \(\dfrac{2x^2 + 3x − 2}{2x^2 + 5x − 3}\)
- \(\dfrac{(t^2 + 4)(2t − 4) − (t^2 − 4t + 4)(2t)}{(t^2 + 4)^2}\)
- \(\dfrac{(2)(x − 4)(x^2 + 4x + 4)}{(x + 2)(x^2 − 16)}\)
- \(\dfrac{12x^2 + 19x − 21}{12x^2 + 38x − 40}\)