Section 1
Simplifying Rational Expressions
When is a fraction or a rational expression undefined?
When does a fraction or a rational expression equal $0 ?$
Evaluate (if possible) for a) $x=3$ and b) $x=-2$.$$\frac{2 x-1}{5 x+2}$$
Evaluate (if possible) for a) $x=3$ and b) $x=-2$.$$\frac{3\left(x^{2}+1\right)}{x^{2}+2 x+1}$$
Evaluate (if possible) for a) $z=1$ and $\mathrm{b}$ ) $z=-3$.$$\frac{(4 z)^{2}}{z^{2}-z-12}$$
Evaluate (if possible) for a) $z=1$ and $\mathrm{b}$ ) $z=-3$.$$\frac{3\left(z^{2}-9\right)}{z^{2}+8}$$
Evaluate (if possible) for a) $z=1$ and $\mathrm{b}$ ) $z=-3$.$$\frac{15+5 z}{16-z^{2}}$$
Evaluate (if possible) for a) $z=1$ and $\mathrm{b}$ ) $z=-3$.$$\frac{4 z-3}{z^{2}+6 z-7}$$
How do you determine the values of the variable for which a rational expression is undefined?
If $x^{2}+9$ is the numerator of a rational expression, can that expression equal zero? Give a reason.
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{m+4}{3 m}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{-y}{y+3}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{2 w-7}{4 w+1}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{3 x+13}{2 x+13}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{11 v-v^{2}}{5 v-9}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$-\frac{r+5}{r^{2}-100}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{8}{p}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{22}{m-1}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$-\frac{7 k}{k^{2}+9 k+20}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{a-9}{a^{2}+8 a-9}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{c+20}{2 c^{2}+3 c-9}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{4}{3 f^{2}-13 f+10}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{g^{2}+9 g+18}{9 g}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{6 m-11}{10}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{4 y}{y^{2}+9}$$
Determine the value(s) of the variable for whicha) the expression equals zero.b) the expression is undefined.$$\frac{q^{2}+49}{7}$$
Write each rational expression in lowest terms.$$\frac{7 x(x-11)}{3(x-11)}$$
Write each rational expression in lowest terms.$$\frac{24(g+3)}{6(g+3)(g-5)}$$
Write each rational expression in lowest terms.$$\frac{24 g^{2}}{56 g^{4}}$$
Write each rational expression in lowest terms.$$\frac{99 d^{7}}{9 d^{3}}$$
Write each rational expression in lowest terms.$$\frac{4 d-20}{5 d-25}$$
Write each rational expression in lowest terms.$$\frac{12 c-3}{8 c-2}$$
Write each rational expression in lowest terms.$$\frac{-14 h-56}{6 h+24}$$
Write each rational expression in lowest terms.$$\frac{-15 v^{2}+12}{40 v^{2}-32}$$
Write each rational expression in lowest terms.$$\frac{39 u^{2}+26}{30 u^{2}+20}$$
Write each rational expression in lowest terms.$$\frac{3 q+15}{-7 q-35}$$
Write each rational expression in lowest terms.$$\frac{g^{2}-g-56}{g+7}$$
Write each rational expression in lowest terms.$$\frac{b^{2}+9 b+20}{b^{2}+b-12}$$
Write each rational expression in lowest terms.$$\frac{t-5}{t^{2}-25}$$
Write each rational expression in lowest terms.$$\frac{r+9}{r^{2}+7 r-18}$$
Write each rational expression in lowest terms.$$\frac{3 c^{2}+28 c+32}{c^{2}+10 c+16}$$
Write each rational expression in lowest terms.$$\frac{6 p^{2}+11 p-10}{9 p^{2}-4}$$
Write each rational expression in lowest terms.$$\frac{q^{2}-25}{2 q^{2}-7 q-15}$$
Write each rational expression in lowest terms.$$\frac{w^{3}+125}{5 w^{2}-25 w+125}$$
Write each rational expression in lowest terms.$$\frac{6 w^{3}-48}{w^{2}+2 w+4}$$
Write each rational expression in lowest terms.$$\frac{9 c^{2}-27 c+81}{c^{3}+27}$$
Write each rational expression in lowest terms.$$\frac{4 x+20}{x^{3}+125}$$
Write each rational expression in lowest terms.$$\frac{4 u^{2}-20 u+4 u v-20 v}{13 u+13 v}$$
Write each rational expression in lowest terms.$$\frac{a b+3 a-6 b-18}{b^{2}-9}$$
Write each rational expression in lowest terms.$$\frac{m^{2}-n^{2}}{m^{3}-n^{3}}$$
Write each rational expression in lowest terms.$$\frac{a^{3}+b^{3}}{a^{2}-b^{2}}$$
Any rational expression of the form $\frac{a-b}{b-a}(a \neq b)$ reduces to what?
Does $\frac{h+4}{h-4}=-1 ?$
Write each rational expression in lowest terms. $$\frac{8-q}{q-8}$$
Write each rational expression in lowest terms.$$\frac{m-15}{15-m}$$
Write each rational expression in lowest terms.$$\frac{m^{2}-121}{11-m}$$
Write each rational expression in lowest terms.$$\frac{k-9}{162-2 k^{2}}$$
Write each rational expression in lowest terms.$$\frac{36-42 x}{7 x^{2}+8 x-12}$$
Write each rational expression in lowest terms.$$\frac{a^{2}-6 a-27}{9-a}$$
Write each rational expression in lowest terms.$$\frac{16-4 b^{2}}{b-2}$$
Write each rational expression in lowest terms.$$\frac{45-9 v}{v^{2}-25}$$
Write each rational expression in lowest terms.$$\frac{y^{3}-3 y^{2}+2 y-6}{21-7 y}$$
Write each rational expression in lowest terms.$$\frac{8 t^{3}-27}{9-4 t^{2}}$$
Write each rational expression in lowest terms.$$\frac{18 c+45}{12 c^{2}+18 c-30}$$
Write each rational expression in lowest terms.$$\frac{36 n^{3}}{42 n^{9}}$$
Write each rational expression in lowest terms.$$\frac{r^{3}-t^{3}}{t^{2}-r^{2}}$$
Write each rational expression in lowest terms.$$\frac{k^{2}-16 k+64}{k^{3}-8 k^{2}+9 k-72}$$
Write each rational expression in lowest terms.$$\frac{b^{2}+6 b-72}{4 b^{2}+52 b+48}$$
Write each rational expression in lowest terms.$$\frac{5 p^{2}-13 p+6}{32-8 p^{2}}$$
Write each rational expression in lowest terms.$$\frac{28 h^{4}-56 h^{3}+7 h}{7 h}$$
Write each rational expression in lowest terms.$$\frac{z^{2}+5 z-36}{64-z^{3}}$$
Write each rational expression in lowest terms.$$\frac{14-6 w}{12 w^{3}-28 w^{2}}$$
Write each rational expression in lowest terms.$$\frac{54 d^{6}+6 d^{5}-42 d^{3}-18 d^{2}}{6 d^{2}}$$
Write each rational expression in lowest terms.$$\frac{-5 v-10}{v^{3}-v^{2}-4 v+4}$$
Write each rational expression in lowest terms.$$\frac{38 x^{2}+38}{-12 x^{2}-12}$$
Find three equivalent forms of each rational expression.$$-\frac{u+7}{u-2}$$
Find three equivalent forms of each rational expression.$$-\frac{8 y-1}{2 y+5}$$
Find three equivalent forms of each rational expression.$$-\frac{9-5 t}{2 t-3}$$
Find three equivalent forms of each rational expression.$$\frac{w-6}{-4 w+7}$$
Find three equivalent forms of each rational expression.$$\frac{-12 m}{m^{2}-3}$$
Find three equivalent forms of each rational expression.$$\frac{-9 x-11}{18-x}$$
Reduce to lowest termsa) using long division.b) using the methods of this section.$$\frac{4 y^{2}-11 y+6}{y-2}$$
Reduce to lowest termsa) using long division.b) using the methods of this section.$$\frac{2 x^{2}+x-28}{x+4}$$
Reduce to lowest termsa) using long division.b) using the methods of this section.$$\frac{8 a^{3}+125}{2 a+5}$$
Reduce to lowest termsa) using long division.b) using the methods of this section.$$\frac{27 t^{3}-8}{3 t-2}$$
Recall that the area of a rectangle is $A=l w,$ where $w=$ width and $l=$ length. Solving for the width, we get $w=\frac{A}{l}$ and solving for the length gives us $l=\frac{A}{w}$Find the missing side in each rectangle.Find the length.
Recall that the area of a rectangle is $A=l w,$ where $w=$ width and $l=$ length. Solving for the width, we get $w=\frac{A}{l}$ and solving for the length gives us $l=\frac{A}{w}$Find the missing side in each rectangle.
Recall that the area of a triangle is $A=\frac{1}{2} b h,$ where $b=$ length of the base and $h=$ height. Solving for the height, we get $h=\frac{2 A}{b}$. Find the height of each triangle.Area $=3 k^{2}+13 k+4$
Recall that the area of a triangle is $A=\frac{1}{2} b h,$ where $b=$ length of the base and $h=$ height. Solving for the height, we get $h=\frac{2 A}{b}$. Find the height of each triangle.Area $=6 p^{2}+52 p+32$
Determine the domain of each rational function.$$f(p)=\frac{1}{p-7}$$
Determine the domain of each rational function.$$h(z)=\frac{z+8}{z+3}$$
Determine the domain of each rational function.$$k(r)=\frac{r}{5 r+2}$$
Determine the domain of each rational function.$$f(a)=\frac{6 a}{7-2 a}$$
Determine the domain of each rational function.$$g(t)=\frac{3 t-4}{t^{2}-9 t+8}$$
Determine the domain of each rational function.$$r(c)=\frac{c+9}{c^{2}-c-42}$$
Determine the domain of each rational function.$$h(w)=\frac{w+7}{w^{2}-81}$$
Determine the domain of each rational function.$$k(t)=\frac{t}{t^{2}-14 t+33}$$
Determine the domain of each rational function.$$A(c)=\frac{8}{c^{2}+6}$$
Determine the domain of each rational function.$$C(n)=\frac{3 n+1}{2}$$
Write your own example of a rational function, $f(x)$, that has a domain of $(-\infty,-8) \cup(-8, \infty)$.
Write your own example of a rational function, $g(x),$ that has a domain of $(-\infty,-5) \cup(-5,6) \cup(6, \infty)$.