3.3.1: Exercises 3.3

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Terms and Concepts

Exercise \(\PageIndex{1}\)

Can the fraction \(\displaystyle \frac{x+2}{x^2+2}\) be simplified? Explain.

Answer: No, the numerator and denominator have no common factors.

Exercise \(\PageIndex{2}\)

In the fraction \(\displaystyle \frac{2}{(x+3)^2(x+2)}\) are there any repeated factors? If so, what factor(s) are repeated, and how many times?

Answer: Yes, \(x+3\) is repeated twice

Exercise \(\PageIndex{3}\)

What is meant by an irreducible quadratic?

Answer: A quadratic that has no real valued roots

Exercise \(\PageIndex{4}\)

Give an example of an irreducible quadratic.

Answer: Answers will vary; \(x^2+a\) is an example if \(a>0\)

Problems

Simplify the given expression in ecercises \(\PageIndex{5}\) - \(\PageIndex{9}\).

Exercise \(\PageIndex{5}\)

\(\displaystyle \frac{5}{18} - \frac{5}{12}\)

Answer: \(\displaystyle \frac{-5}{36}\)

Exercise \(\PageIndex{6}\)

\(\displaystyle \frac{x}{b} - \frac{b}{x}\)

Answer: \(\displaystyle \frac{x^2-b^2}{xb}\)

Exercise \(\PageIndex{7}\)

\(\displaystyle \frac{x}{y^2} - \frac{x}{x+y}\)

Answer: \(\displaystyle \frac{x^2+xy-xy^2}{xy^2 + y^3}\)

Exercise \(\PageIndex{8}\)

\(\displaystyle \frac{\phantom{x} \frac{1}{x} - \frac{x+2}{x^2} \phantom{x}}{\frac{4}{x^2} - \frac{x^2+1}{x^3}}\)

Answer: \(\displaystyle \frac{\phantom{x} -2x}{-x^2+4x-1 \phantom{x} }\), \(x \neq 0\)

Exercise \(\PageIndex{9}\)

\(\displaystyle \frac{\phantom{x} \frac{1}{x-b} - \frac{1}{x} \phantom{x}}{b}\)

Answer: \(\displaystyle \frac{1}{x^2-bx}\), \(b\neq 0\)

In exercises \(\PageIndex{10}\) - \(\PageIndex{16}\), decompose the given fraction. Do not solve for \(A\), \(B\), etc.

Exercise \(\PageIndex{10}\)

\(\displaystyle \frac{x-8}{(x+2)^3}\)

Answer: \(\displaystyle \frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3}\)

Exercise \(\PageIndex{11}\)

\(\displaystyle \frac{4}{(s-1)^2(2s-5)(s+3)}\)

Answer: \(\displaystyle \frac{A}{s-1} + \frac{B}{(s-1)^2} + \frac{C}{2s-5} + \frac{D}{s+3}\)

Exercise \(\PageIndex{12}\)

\(\displaystyle \frac{5t^2+11t-9}{(t+1)^3(t^2+1)^2}\)

Answer: \(\displaystyle \frac{A}{t+1} + \frac{B}{(t+1)^2} + \frac{C}{(t+1)^3} + \frac{Dt+E}{t^2+1} + \frac{Ft+G}{(t^2+1)^2}\)

Exercise \(\PageIndex{13}\)

\(\displaystyle \frac{6x}{(x-4)(x^2+x+5)}\)

Answer: \(\displaystyle \frac{A}{x-4} + \frac{Bx+C}{x^2+x+5}\)

Exercise \(\PageIndex{14}\)

\(\displaystyle \frac{3x-7}{x^4-1}\)

Answer: \(\displaystyle \frac{A}{x+1} + \frac{B}{x-1} + \frac{Cx+D}{x^2+1}\)

Exercise \(\PageIndex{15}\)

\(\displaystyle \frac{2s}{s^3+1}\)

Answer: \(\displaystyle \frac{A}{s+1} + \frac{Bs+C}{s^2-s+1}\)

Exercise \(\PageIndex{16}\)

\(\displaystyle \frac{11}{t^2-6t+5}\)

Answer: \(\displaystyle \frac{A}{t-5} + \frac{B}{t-1}\)

In exercises \(\PageIndex{17}\) - \(\PageIndex{22}\), fully decompose the given fraction.

Exercise \(\PageIndex{17}\)

\(\displaystyle \frac{x+5}{x^2+x-2}\)

Answer: \(\displaystyle \frac{-1}{x+2} + \frac{2}{x-1}\)

Exercise \(\PageIndex{18}\)

\(\displaystyle \frac{1}{x^2-a^2}\)

Answer: \(\displaystyle \frac{1/(2a)}{x-a} - \frac{1/(2a)}{x+a}\)

Exercise \(\PageIndex{19}\)

\(\displaystyle \frac{2s^2-s+4}{s^3+4s}\)

Answer: \(\displaystyle \frac{1}{s} + \frac{s-1}{s^2+4}\)

Exercise \(\PageIndex{20}\)

\(\displaystyle \frac{y-1}{y^2+3y+2}\)

Answer: \(\displaystyle \frac{3}{y+2} - \frac{2}{y+1}\)

Exercise \(\PageIndex{21}\)

\(\displaystyle \frac{4x}{x^3-x^2-x+1}\)

Answer: \(\displaystyle \frac{-1}{x+1} + \frac{1}{x-1} + \frac{2}{(x-1)^2}\)

Exercise \(\PageIndex{22}\)

\(\displaystyle \frac{x^2+2x-1}{2x^3+3x^2-2x}\)

Answer: \(\displaystyle \frac{1/2}{x} + \frac{1/5}{2x-1} - \frac{1/10}{x+2}\)