8.1 Simplify rational expressions  (Page 3/5)

Solution

Rewrite the numerator and denominator showing the common factors.
Simplify using the Equivalent Fractions Property.

Did you notice that these are the same steps we took when we divided monomials in Polynomials ?

Solution

Solution

$\begin{array}{cccc}& & & \hfill \frac{x^2+5x+6}{x^2+8x+12}\hfill \\ \text{Factor the numerator and denominator.}\hfill & & & \hfill \frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+6\right)}\hfill \\ \\ \\ \begin{array}{c}\text{Remove the common factor}x+2\text{from}\hfill \\ \text{the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \frac{\cancel{\left(x+2\right)}\left(x+3\right)}{\cancel{\left(x+2\right)}\left(x+6\right)}\hfill \\ & & & \hfill \frac{x+3}{x+6}\hfill \end{array}$

Can you tell which values of x must be excluded in this example?

Solution

$\begin{array}{cccc}& & & \hfill \frac{y^2+y-42}{y^2-36}\hfill \\ \text{Factor the numerator and denominator.}\hfill & & & \hfill \frac{\left(y+7\right)\left(y-6\right)}{\left(y+6\right)\left(y-6\right)}\hfill \\ \\ \\ \begin{array}{c}\text{Remove the common factor}y-6\text{from}\hfill \\ \text{the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \frac{\left(y+7\right)\cancel{\left(y-6\right)}}{\left(y+6\right)\cancel{\left(y-6\right)}}\hfill \\ & & & \hfill \frac{y+7}{y+6}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{p^3-2p^2+2p-4}{p^2-7p+10}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{using grouping to factor the numerator.}\hfill \end{array}\hfill & & & \hfill \frac{p^2\left(p-2\right)+2\left(p-2\right)}{\left(p-5\right)\left(p-2\right)}\hfill \\ & & & \hfill \frac{\left(p^2+2\right)\left(p-2\right)}{\left(p-5\right)\left(p-2\right)}\hfill \\ \\ \\ \begin{array}{c}\text{Remove the common factor of}p-2\hfill \\ \text{from the numerator and the denominator.}\hfill \end{array}\hfill & & & \hfill \frac{\left(p^2+2\right)\cancel{\left(p-2\right)}}{\left(p-5\right)\cancel{\left(p-2\right)}}\hfill \\ & & & \hfill \frac{p^2+2}{p-5}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{2n^2-14n}{4n^2-16n-48}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{first factoring out the GCF.}\hfill \end{array}\hfill & & & \hfill \frac{2n\left(n-7\right)}{4\left(n^2-4n-12\right)}\hfill \\ & & & \hfill \frac{2n\left(n-7\right)}{4\left(n-6\right)\left(n+2\right)}\hfill \\ \\ \\ \text{Remove the common factor, 2.}\hfill & & & \hfill \frac{\cancel{2}n\left(n-7\right)}{\cancel{2}·2\left(n-6\right)\left(n+2\right)}\hfill \\ & & & \hfill \frac{n\left(n-7\right)}{2\left(n-6\right)\left(n+2\right)}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{3b^2-12b+12}{6b^2-24}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{first factoring out the GCF.}\hfill \end{array}\hfill & & & \hfill \frac{3\left(b^2-4b+4\right)}{6\left(b^2-4\right)}\hfill \\ & & & \hfill \frac{3\left(b-2\right)\left(b-2\right)}{6\left(b+2\right)\left(b-2\right)}\hfill \\ \\ \\ \text{Remove the common factors of}b-2\text{and}3.\hfill & & & \hfill \frac{\cancel{3}\left(b-2\right)\cancel{\left(b-2\right)}}{\cancel{3}·2\left(b+2\right)\cancel{\left(b-2\right)}}\hfill \\ & & & \hfill \frac{b-2}{2\left(b+2\right)}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{m^3+8}{m^2-4}\hfill \\ \begin{array}{c}\text{Factor the numerator and denominator,}\hfill \\ \text{using the formulas for sum of cubes and}\hfill \\ \text{difference of squares.}\hfill \end{array}\hfill & & & \hfill \frac{\left(m+2\right)\left(m^2-2m+4\right)}{\left(m+2\right)\left(m-2\right)}\hfill \\ \text{Remove the common factor of}m+2.\hfill & & & \hfill \frac{\cancel{\left(m+2\right)}\left(m^2-2m+4\right)}{\cancel{\left(m+2\right)}\left(m-2\right)}\hfill \\ & & & \hfill \frac{m^2-2m+4}{m-2}\hfill \end{array}$