# 3.3.1: Exercises 3.3

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Terms and Concepts

Exercise $$\PageIndex{1}$$

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

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?

Yes, $$x+3$$ is repeated twice

Exercise $$\PageIndex{3}$$

What is meant by an irreducible quadratic?

A quadratic that has no real valued roots

Exercise $$\PageIndex{4}$$

Give an example of an irreducible quadratic.

Answers will vary; $$x^2+a$$ is an example if $$a>0$$

## Problems

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

Exercise $$\PageIndex{5}$$

$$\displaystyle \frac{5}{18} - \frac{5}{12}$$

$$\displaystyle \frac{-5}{36}$$

Exercise $$\PageIndex{6}$$

$$\displaystyle \frac{x}{b} - \frac{b}{x}$$

$$\displaystyle \frac{x^2-b^2}{xb}$$

Exercise $$\PageIndex{7}$$

$$\displaystyle \frac{x}{y^2} - \frac{x}{x+y}$$

$$\displaystyle \frac{x^2+xy-xy^2}{xy^2 + y^3}$$

Exercise $$\PageIndex{8}$$

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

$$\displaystyle \frac{\phantom{x} -2x}{-x^2+4x-1 \phantom{x} }$$, $$x \neq 0$$

Exercise $$\PageIndex{9}$$

$$\displaystyle \frac{\phantom{x} \frac{1}{x-b} - \frac{1}{x} \phantom{x}}{b}$$

$$\displaystyle \frac{1}{x^2-bx}$$, $$b\neq 0$$

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

Exercise $$\PageIndex{10}$$

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

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

Exercise $$\PageIndex{11}$$

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

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

Exercise $$\PageIndex{12}$$

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

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

Exercise $$\PageIndex{13}$$

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

$$\displaystyle \frac{A}{x-4} + \frac{Bx+C}{x^2+x+5}$$

Exercise $$\PageIndex{14}$$

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

$$\displaystyle \frac{A}{x+1} + \frac{B}{x-1} + \frac{Cx+D}{x^2+1}$$

Exercise $$\PageIndex{15}$$

$$\displaystyle \frac{2s}{s^3+1}$$

$$\displaystyle \frac{A}{s+1} + \frac{Bs+C}{s^2-s+1}$$

Exercise $$\PageIndex{16}$$

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

$$\displaystyle \frac{A}{t-5} + \frac{B}{t-1}$$

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

Exercise $$\PageIndex{17}$$

$$\displaystyle \frac{x+5}{x^2+x-2}$$

$$\displaystyle \frac{-1}{x+2} + \frac{2}{x-1}$$

Exercise $$\PageIndex{18}$$

$$\displaystyle \frac{1}{x^2-a^2}$$

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

Exercise $$\PageIndex{19}$$

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

$$\displaystyle \frac{1}{s} + \frac{s-1}{s^2+4}$$

Exercise $$\PageIndex{20}$$

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

$$\displaystyle \frac{3}{y+2} - \frac{2}{y+1}$$

Exercise $$\PageIndex{21}$$

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

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

Exercise $$\PageIndex{22}$$

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

$$\displaystyle \frac{1/2}{x} + \frac{1/5}{2x-1} - \frac{1/10}{x+2}$$