Section 1
Simplifying Rational Expressions
Fill in the blanks.A quotient of two polynomials, such as $\frac{x^{2}+x}{x^{2}-3 x},$ is called a ____ expression.
Fill in the blanks.To simplify a rational expression, we remove common ____ of the numerator and denominator.
Fill in the blanks.Because of the division by $0,$ the expression $\frac{8}{0}$ is ____.
Fill in the blanks.The binomials $x-15$ and $15-x$ are called _____ because their terms are the same, except that they are opposite in sign.
When we simplify $\frac{x^{2}+5 x}{4 x+20},$ the result is $\frac{x}{4} .$ These equivalent expressions have the same value for all real numbers, except $x=-5 .$ Show that they have the same value for $x=1$
Determine whether each pair of polynomials are opposites. Write yes or no.a. $y+7$ and $y-7$b. $b-20$ and $20-b$c. $x^{2}+2 x-1$ and $-x^{2}-2 x-1$
Simplify each expression, if possible.a. $\frac{x-8}{x-8}$b. $\frac{x-8}{8-x}$c. $\frac{x+8}{8+x}$d. $\frac{x+8}{x}$
Simplify each expression.a. $\frac{(x+2)(x-2)}{(x+1)(x+2)}$b. $\frac{y(y-2)}{9(2-y)}$c. $\frac{(2 m+7)(m-5)}{(2 m+7)}$d. $\frac{x \cdot x}{x \cdot x(x-30)}$
Complete the solution to simplify the rational expression.$$\begin{aligned}\frac{x^{2}+2 x+1}{x^{2}+4 x+3} &=\frac{(x+1)(1+1)}{(x+3)(x+1)} \\&=\frac{(x+1)(x+1)}{(x+3)} \\&=x+1\end{aligned}$$
In the following table, a student's answers to three homework problems are compared with the answers in the back of the book. Are the answers equivalent?(TABLE NOT COPY)
Evaluate each expression for $x=6 .$ See Example 1.$$\frac{x-2}{x-5}$$
Evaluate each expression for $x=6 .$ See Example 1.$$\frac{3 x-2}{x-2}$$
Evaluate each expression for $x=6 .$ See Example 1.$$\frac{x^{2}-4 x-12}{x^{2}+x-2}$$
Evaluate each expression for $x=6 .$ See Example 1.$$\frac{x^{2}-36}{x^{3}-1}$$
Evaluate each expression for $x=6 .$ See Example 1.$$\frac{-x+1}{x^{2}-5 x-6}$$
Evaluate each expression for $x=6 .$ See Example 1.$$\frac{-2 x^{2}-3}{x-6}$$
Evaluate each expression for $y=-3 .$ See Example 1.$$\frac{y+5}{3 y-2}$$
Evaluate each expression for $y=-3 .$ See Example 1.$$\frac{2 y+9}{y^{2}+25}$$
Evaluate each expression for $y=-3 .$ See Example 1.$$-\frac{y}{y^{2}-y+6}$$
Evaluate each expression for $y=-3 .$ See Example 1.$$-\frac{y^{3}}{3 y^{2}+1}$$
Evaluate each expression for $y=-3 .$ See Example 1.$$\frac{y^{2}+9}{9-y^{2}}$$
Evaluate each expression for $y=-3 .$ See Example 1.$$\frac{-y-11}{y^{2}+2 y-3}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{15}{x-2}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{5 x}{x+5}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{x+5}{8 x}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{4 x-1}{6 x}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{15 x+2}{x^{2}+6}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{x^{2}-4 x}{x^{2}+4}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{x+1}{2 x-1}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{-6 x}{3 x-1}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{x^{2}-6 x}{9}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{x^{3}-x^{2}}{15}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{30 x}{x^{2}-36}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{2 x-15}{x^{2}-49}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{15}{x^{2}+x-2}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{x-20}{x^{2}+2 x-8}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{16}{20-x}$$
Find all real numbers for which the rational expression is undefined. See Example 2.$$\frac{44}{57-x}$$
Simplify. See Example 3.$$\frac{45}{9 a}$$
Simplify. See Example 3.$$\frac{48}{16 y}$$
Simplify. See Example 3.$$\frac{6 x^{4}}{4 x^{2}}$$
Simplify. See Example 3.$$\frac{9 x^{3}}{6 x}$$
Simplify. See Example 4.$$\frac{6 x+3}{9}$$
Simplify. See Example 4.$$\frac{4 x+12}{16}$$
Simplify. See Example 4.$$\frac{x+3}{3 x+9}$$
Simplify. See Example 4.$$\frac{2 x-14}{x-7}$$
Simplify. See Example 4.$$\frac{x^{2}-4}{x^{2}-6 x+8}$$
Simplify. See Example 4.$$\frac{y^{2}-25}{y^{2}-3 y-10}$$
Simplify. See Example 4.$$\frac{x^{2}+5 x+4}{x^{2}+4 x}$$
Simplify. See Example 4.$$\frac{x^{2}-10 x+21}{x^{2}-3 x}$$
Simplify. See Example $5 .$$$\frac{m^{2}-2 m n+n^{2}}{7 m^{2}-7 n^{2}}$$
Simplify. See Example $5 .$$$\frac{11 c^{2}-11 d^{2}}{c^{2}-2 c d+d^{2}}$$
Simplify. See Example $5 .$$$\frac{4 b^{2}+4 b+1}{(2 b+1)^{3}}$$
Simplify. See Example $5 .$$$\frac{9 y^{2}-12 y+4}{(3 y-2)^{3}}$$
Simplify. See Example $6 .$$$\frac{10(c-3)+10}{3(c-3)+3}$$
Simplify. See Example $6 .$$$\frac{6(d+3)-6}{7(d+3)-7}$$
Simplify. See Example $6 .$$$\frac{6(x+3)-18}{3 x-18}$$
Simplify. See Example $6 .$$$\frac{4(t-1)+4}{4 t+4}$$
Simplify. See Example 7 .$$\frac{2 x-7}{7-2 x}$$
Simplify. See Example 7 .$$\frac{18-d}{d-18}$$
Simplify. See Example 7 .$$\frac{3-4 t}{8 t-6}$$
Simplify. See Example 7 .$$\frac{5 t-1}{3-15 t}$$
Simplify. See Example $8 .$$$\frac{2-a}{a^{2}-a-2}$$
Simplify. See Example $8 .$$$\frac{4-b}{b^{2}-5 b+4}$$
Simplify. See Example $8 .$$$\frac{25-5 m}{m^{2}-25}$$
Simplify. See Example $8 .$$$\frac{36-6 h}{h^{2}-36}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{a^{3}-a^{2}}{a^{4}-a^{3}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{2 c^{4}+2 c^{3}}{4 c^{5}+4 c^{4}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{4-x^{2}}{x^{2}-x-2}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{81-y^{2}}{y^{2}+10 y+9}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{6 x-30}{5-x}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{6 t-42}{7-t}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{x^{2}+3 x+2}{x^{2}+x-2}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{x^{2}+x-6}{x^{2}-x-2}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{15 x^{2} y}{5 x y^{2}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{12 x z}{4 x z^{2}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{x^{8}+9 x^{7}}{9+x}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{x^{9}+50 x^{8}}{50+x}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{x(x-8)+16}{16-x^{2}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{x^{2}-3(2 x-3)}{9-x^{2}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{4 c+4 d}{d+c}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{a+b}{5 b+5 a}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{3 x^{2}-27}{2 x^{2}-5 x-3}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{2 x^{2}-8}{3 x^{2}-5 x-2}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{-3 x^{2}+10 x+77}{x^{2}-4 x-21}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{-2 x^{2}+5 x+3}{x^{2}+2 x-15}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{42 c^{3} d}{18 c d^{3}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{49 m^{4} n^{5}}{35 m n^{6}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{16 a^{2}-1}{4 a+4}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{25 m^{2}-1}{5 m+5}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{8 u^{2}-2 u-15}{4 u^{4}+5 u^{3}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{6 n^{2}-7 n+2}{3 n^{3}-2 n^{2}}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{(2 x+3)^{4}}{4 x^{2}+12 x+9}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{(3 y-2)^{5}}{9 y^{2}-12 y+4}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{6 a+3(a+2)+12}{a+2}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{2 y+4(y-1)-2}{y-1}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{15 x-3 x^{2}}{25 y-5 x y}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{18 c-2 c^{2}}{81 d-9 c d}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{2 x^{2}}{x+2}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{5 y^{2}}{y+5}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{18+2 x}{x^{2}-81}$$
Simplify. If an expression cannot be simplified, write "Does not simplify."$$\frac{12+6 x}{x^{2}-4}$$
The number of vibrations $n$ per second of an organ pipe is given by the formula $n=\frac{512}{L}$ where $L$ is the length of the pipe in feet. How many times per second will a 6 -foot pipe vibrate?
Raising Turkeys. The formula $T=\frac{2,000 m}{m+1}$ gives the number $T$ of turkeys on a poultry farm $m$ months after the beginning of the year. How many turkeys will there be on the farm by the end of July?
The formula $c=\frac{4 t}{t^{2}+1}$ gives the concentration $c$ (in milligrams per liter) of a certain dosage of medication in a patient's bloodstream $t$ hours after the medication is administered. Suppose the patient received the medication at noon. Find the concentration of medication in his blood at the following times later that afternoon.(PICTURE NOT COPY)
If a company produces $x$ child car seats, the average cost $c$ (in dollars) to produce one car seat is given by the formula $c=\frac{50 x+50,000}{x} .$ Find the company's average production cost if $1,000$ are produced.
Explain why $\frac{x-7}{7-x}=-1$
Explain why $\frac{x-3}{x+4}$ is undefined for $x=-4$ but defined for $x=3$
Explain the error in the following work:Simplify: (EQUATION NOT COPY)
Explain why there are no values for $x$ for which $\frac{x-7}{x^{2}+49}$ is undefined.
Write a rational expression that is not defined for $x=5 .$ Then explain why that is so.
State each property using the variables $a, b,$ and when necessary, c.a. The associative property of additionb. The commutative property of multiplication
State each property using the variables $a, b,$ and when necessary, c.a. The distributive propertyb. The zero-factor property
Simplify.$$\frac{\left(x^{2}+2 x+1\right)\left(x^{2}-2 x+1\right)}{\left(x^{2}-1\right)^{2}}$$
Simplify.$$\frac{2 x^{2}+2 x-12}{x^{3}+3 x^{2}-4 x-12}$$
Simplify.$$\frac{x^{3}-27}{x^{3}-9 x}$$
Simplify.$$\frac{b^{3}+a^{3}}{a^{2}-a b+b^{2}}$$
Simplify.$$\frac{m^{3}+64}{m^{3}+4 m^{2}+3 m+12}$$
Simplify.$$\frac{s^{3}+s^{2}-6 s-6}{s^{3}+1}$$