Chapter 4

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{4 x}{x-5}-3 $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{2 y-2 x y}{x^{2} y-y}\)

How would you help someone solve the equation $$ \frac{3}{x}-\frac{4}{x}=\frac{-1}{x} \text { ? } $$

Use synthetic division to determine the quotient and remainder. $$ \left(2 x^{4}+3 x^{2}+3\right) \div(x+2) $$

Simplify each complex fraction. $$ 1+\frac{x}{1+\frac{1}{x}} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7 x}{x+4}-2 $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{3 x-x^{2}}{x^{2}-9}\)

Describe the process of long division of polynomials.

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ -1-\frac{3}{2 x+1} $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{2 x^{3}-8 x}{4 x-x^{3}}\)

Give a step-by-step description of how you would do the following division problem. $$ \left(4-3 x-7 x^{3}\right) \div(x+6) $$

Give a step-by-step description of how to do the following addition problem. $$ \frac{3 x+4}{8}+\frac{5 x-2}{12} $$

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ -2-\frac{5}{4 x-3} $$

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{x^{2}-(y-1)^{2}}{(y-1)^{2}-x^{2}}\)

How do you know by inspection that \(3 x^{2}+5 x+1\) cannot be the correct answer for the division problem \(\left(3 x^{3}-7 x^{2}-22 x+8\right) \div(x-4) ?\)

Recall that the indicated quotient of a polynomial and its opposite is \(-1\). For example, \(\frac{x-2}{2-x}\) simplifies to \(-1\). Keep this idea in mind as you add or subtract the following rational expressions. (a) \(\frac{1}{x-1}-\frac{x}{x-1}\) (b) \(\frac{3}{2 x-3}-\frac{2 x}{2 x-3}\) (c) \(\frac{4}{x-4}-\frac{x}{x-4}+1\) (d) \(-1+\frac{2}{x-2}-\frac{x}{x-2}\)

For Problems 59-68, simplify each rational expression. You may want to refer to Example 12 of this section. \(\frac{n^{2}-5 n-24}{40+3 n-n^{2}}\)

Consider the addition problem \(\frac{8}{x-2}+\frac{5}{2-x}\). Note that the denominators are opposites of each other. If the property \(\frac{a}{-b}=-\frac{a}{b}\) is applied to the second fraction, we have \(\frac{5}{2-x}=-\frac{5}{x-2}\). Thus we proceed as follows: $$ \frac{8}{x-2}+\frac{5}{2-x}=\frac{8}{x-2}-\frac{5}{x-2}=\frac{8-5}{x-2}=\frac{3}{x-2} $$ Use this approach to do the following problems. (a) \(\frac{7}{x-1}+\frac{2}{1-x}\) (b) \(\frac{5}{2 x-1}+\frac{8}{1-2 x}\) (c) \(\frac{4}{a-3}-\frac{1}{3-a}\) (d) \(\frac{10}{a-9}-\frac{5}{9-a}\) (e) \(\frac{x^{2}}{x-1}-\frac{2 x-3}{1-x}\) (f) \(\frac{x^{2}}{x-4}-\frac{3 x-28}{4-x}\)

What is the difference between the concept of least common multiple and the concept of least common denominator?

Compare the concept of a rational number in arithmetic to the concept of a rational expression in algebra.

A classmate tells you that she finds the least common multiple of two counting numbers by listing the multiples of each number and then choosing the smallest number that appears in both lists. Is this a correct procedure? What is the weakness of this procedure?

What role does factoring play in the simplifying of rational expressions?

For which real numbers does \(\frac{x}{x-3}+\frac{4}{x}\) equal \(\frac{(x+6)(x-2)}{x(x-3)} ?\) Explain your answer.

Why is the rational expression \(\frac{x+3}{x^{2}-4}\) undefined for \(x=2\) and \(x=-2\) but defined for \(x=-3 ?\)

Suppose that your friend does an addition problem as follows: $$ \frac{5}{8}+\frac{7}{12}=\frac{5(12)+8(7)}{8(12)}=\frac{60+56}{96}=\frac{116}{96}=\frac{29}{24} $$ Is this answer correct? If not, what advice would you offer your friend?

How would you convince someone that \(\frac{x-4}{4-x}=-1\) for all real numbers except 4 ?