유리함수

$\frac{8x}{y}$

$\frac{3}{2}$

$\frac{7}{5}$

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

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

$\frac{x+1}{x-1}$

$\frac{x-5}{x-1}$

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

$\frac{x+3}{x+2}$

$\frac{x+1}{x-7}$

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

$\frac{{y}^{2}+1}{y+2}$

$\frac{{p}^{2}+2}{p+5}$

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

$\frac{n}{2(n+1)}$

$\frac{x}{2(x+2)}$

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

$\frac{2(x-3)}{3(x+3)}$

$\frac{5(y-1)}{2(y+5)}$

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