Arithmetic of Rational Expressions: We have seen that, just like numbers, polynomials have their own arithmetic (addition, subtraction, multiplication, and division).
We shall see that rational expressions have their own arithmetic as well.
Fact: In this section we begin with multiplication and division.
Question: Why?
Answer: Because multiplication and division are easiest!
Multiplying Rational Expressions
Recall: remember how we multiply two fractions? Of course you do! $$\displaystyle \frac{a}{b} \cdot \displaystyle \frac{c}{d}=\displaystyle \frac{ac}{bd}$$ Great News! The SAME RULE APPLIES to rational expressions.
Example: $\displaystyle \frac{2x^4}{-3} \cdot \displaystyle \frac{-20}{-36x^3}$
$$
\begin{array}{cll}
\displaystyle \frac{2x^4}{-3} \cdot \displaystyle \frac{-20}{-36x^3} &= \displaystyle \frac{(2x^4)(-20)}{-3(-36x^3)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle -\frac{(2x^4)(20)}{3(36x^3)} &\mbox{odd number of negatives means overall sign is negative}\\
&= \displaystyle -\frac{2\cdot x \cdot x\cdot x\cdot x \cdot 2 \cdot 2 \cdot 5}{3 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot x\cdot x\cdot x} &\mbox{factor numerators and denominators}\\
&= \displaystyle -\frac{\color{magenta}{2}\cdot \color{magenta}{2} \cdot 2 \cdot 5 \cdot x \cdot \color{magenta}{x}\cdot \color{magenta}{x}\cdot \color{magenta}{x} }{\color{magenta}{2} \cdot \color{magenta}{2} \cdot 3 \cdot 3 \cdot 3 \cdot \color{magenta}{x}\cdot \color{magenta}{x}\cdot \color{magenta}{x}} &\mbox{rearrange factors and identify }\color{magenta}{\mbox{common}} \mbox{ factors}\\
&= \displaystyle -\frac{ 2 \cdot 5 \cdot x }{ 3 \cdot 3 \cdot 3 } &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\
&= \displaystyle -\frac{ 10x }{27 } &\mbox{simplify}\\
\end{array}
$$
Multiplying Rational Expressions
Process: To multiply two rational expressions:
$1)$ Write the product of the numerators and denominators.
$2)$ Factor both numerators and denominators.
$3)$ Cancel any common factors.
$4)$ If necessary, simplify.
Multiplying Rational Expressions
Examples
$\displaystyle \frac{t^2-121 a^2}{3t^2-34t a+11 a^2} \cdot \displaystyle \frac{9t^2-6t a+a^2}{9t^2- a^2}$
$$
\begin{array}{cll}
\displaystyle \frac{t^2-121 a^2}{3t^2-34t a+11 a^2} \cdot \displaystyle \frac{9t^2-6t a+a^2}{9t^2- a^2} &= \displaystyle \frac{(t^2-121 a^2)(9t^2-6t a+a^2)}{(3t^2-34t a+11 a^2)(9t^2- a^2)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{(t+11a)\color{magenta}{(t-11a)}\color{magenta}{(3t-a)}\color{magenta}{(3t-a)}}{\color{magenta}{(3t-a)}\color{magenta}{(t-11a)}(3t+a)\color{magenta}{(3t-a)}} &\mbox{factor numerators and denominators}\\
&= \displaystyle \frac{ t+11a}{ 3t+a} &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\
\end{array}
$$
$\displaystyle \frac{121 t^2-121 t s+121 s^2}{7 t s} \cdot \displaystyle \frac{5 t^2 s}{11 t^2-11 t s+11 s^2}$
$$
\begin{array}{cll}
\displaystyle \frac{121 t^2-121 t s+121 s^2}{7 t s} \cdot \displaystyle \frac{5 t^2 s}{11 t^2-11 t s+11 s^2} &= \displaystyle \frac{(121 t^2-121 t s+121 s^2)(5 t^2 s)}{(7 t s)(11 t^2-11 t s+11 s^2)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{\color{magenta}{11}\cdot 11 \cdot \color{magenta}{(t^2-st+s^2)}\cdot 5 \cdot t \cdot \color{magenta}{t} \cdot \color{magenta}{s}}{7 \cdot \color{magenta}{t} \cdot \color{magenta}{s} \cdot \color{magenta}{11} \cdot \color{magenta}{(t^2-st+s^2)}} &\mbox{factor numerators and denominators}\\
&= \displaystyle \frac{11\cdot 5 \cdot t }{7} &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\
&= \displaystyle \frac{55t }{7} &\mbox{simplify}\\
\end{array}
$$
$(3 n^2-11 n -4) \cdot \displaystyle \frac{n^2 -3 n-28}{3 n^2-20 n -7}$
$$
\begin{array}{cll}
\displaystyle (3 n^2-11 n -4) \cdot \displaystyle \frac{n^2 -3 n-28}{3 n^2-20 n -7} &= \displaystyle \frac{(3 n^2-11 n -4)}{1} \cdot \displaystyle \frac{n^2 -3 n-28}{3 n^2-20 n -7} &\mbox{write both in fraction form}\\
&= \displaystyle \frac{(3 n^2-11 n -4)(n^2 -3 n-28)}{1\cdot(3 n^2-20 n -7)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{\color{magenta}{(3n+1)}(n-4)(n+4)\color{magenta}{(n-7)}}{1\cdot\color{magenta}{(3n+1)(n-7)}} &\mbox{factor numerators and denominators}\\
&= \displaystyle \frac{(n-4)(n+4)}{1} &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\
&= \displaystyle (n-4)(n+4) &\mbox{simplify}\\
\end{array}
$$
$\displaystyle \frac{176 t^2 r+132 t r^2}{16 t^2-9 r^2} \cdot \displaystyle \frac{4 t-3 r}{44 t r}$
$$
\begin{array}{cll}
\displaystyle \frac{176 t^2 r+132 t r^2}{16 t^2-9 r^2} \cdot \displaystyle \frac{4 t-3 r}{44 t r} &= \displaystyle \frac{(176 t^2 r+132 t r^2)(4 t-3 r)}{(16 t^2-9 r^2)(44 t r)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{(\color{darkorange}{2}\cdot \color{darkorange}{2}\cdot 2 \cdot 2 \cdot \color{darkorange}{11}\cdot t\cdot \color{darkorange}{t} \cdot \color{darkorange}{r}+ \color{darkorange}{2}\cdot \color{darkorange}{2} \cdot 3 \cdot \color{darkorange}{11} \cdot \color{darkorange}{t} \cdot \color{darkorange}{r} \cdot r)(4 t-3 r)}{(16 t^2-9 r^2)(44 t r)} &\mbox{determine } \color{darkorange}{\mbox{GCF}} \mbox{ to factor numerator}\\
&= \displaystyle \frac{\color{darkorange}{2\cdot 2 \cdot 11 \cdot t \cdot r}\cdot (2 \cdot 2 \cdot t+ 3 \cdot r)(4 t-3 r)}{(16 t^2-9 r^2)(44 t r)} &\mbox{factor out } \color{darkorange}{\mbox{GCF}} \\
&= \displaystyle \frac{\color{darkorange}{2\cdot 2 \cdot 11 \cdot t \cdot r}\cdot (4t+ 3 r)(4 t-3 r)}{(16 t^2-9 r^2)(44 t r)} &\mbox{simplify} \\
&= \displaystyle \frac{2\cdot 2 \cdot 11 \cdot t \cdot r \cdot (4t+3r)(4t-3r)}{(4t+3r)(4t-3r)\cdot 2 \cdot 2 \cdot 11 \cdot t \cdot r} &\mbox{factor all other numerators and denominators}\\
&= \displaystyle \frac{\color{magenta}{2}\cdot \color{magenta}{2} \cdot \color{magenta}{11} \cdot \color{magenta}{t} \cdot \color{magenta}{r} \cdot \color{magenta}{(4t+3r)}\color{magenta}{(4t-3r)}}{\color{magenta}{(4t+3r)}\color{magenta}{(4t-3r)}\cdot \color{magenta}{2} \cdot \color{magenta}{2} \cdot \color{magenta}{11} \cdot \color{magenta}{t} \cdot \color{magenta}{r}} &\mbox{identify }\color{magenta}{\mbox{common}} \mbox{ factors}\\\\
&= \displaystyle \frac{1}{1} &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\\\
&= \displaystyle 1 &\mbox{simplify}\\\\
\end{array}
$$
There's always a $1$ left over if everything cancels on either the top or bottom.
Dividing Rational Expressions
Recall: fraction division. $$\displaystyle \frac{a}{b} \div \displaystyle \frac{c}{d}=\displaystyle \frac{a}{b} \cdot \displaystyle \frac{d}{c}=\displaystyle \frac{ad}{bc}$$ Great News! The SAME RULE APPLIES to rational expressions.
Example: $\displaystyle \frac{x^2-15x+44}{x^2-8x+16} \div \displaystyle \frac{x^2-22x+121}{7 x^2-112}$
$$
\begin{array}{cll}
\displaystyle \frac{x^2-15x+44}{x^2-8x+16} \div \displaystyle \frac{\color{blue}{x^2-22x+121}}{\color{red}{7 x^2-112}} &= \displaystyle \frac{x^2-15x+44}{x^2-8x+16} \cdot \displaystyle \frac{\color{red}{7 x^2-112}}{\color{blue}{x^2-22x+121}} &\mbox{invert and multiply}\\
&= \displaystyle \frac{(x^2-15x+44)(7 x^2-112)}{(x^2-8x+16)(x^2-22x+121)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{\color{magenta}{(x-11)}\color{magenta}{(x-4)}\cdot 7 \cdot (x+4)\color{magenta}{(x-4)}}{\color{magenta}{(x-4)(x-4)}\color{magenta}{(x-11)}(x-11)} &\mbox{factor numerators and denominators}\\
&= \displaystyle \frac{7(x+4)}{x-11} &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\
&= \displaystyle \frac{7x+28}{x-11} &\mbox{simplify}\\
\end{array}
$$
Dividing Rational Expressions
Process: To divide two rational expressions:
$0)$ Rewrite write the division problem as a multiplication problem (invert and multiply).
$1)$ Write the product of the numerators and denominators.
$2)$ Factor both numerators and denominators.
$3)$ Cancel any common factors.
$4)$ If necessary, simplify.
Note: steps $1,$ $2,$ $3,$ and $4$ above are the same steps for multiplication with only the additional "invert and multiply" step.
Dividing Rational Expressions
Examples
$\displaystyle \frac{p^2-y^2}{p^2-2p y+y^2} \div \displaystyle \frac{3 p+3 y}{21p-12}$
$$
\begin{array}{cll}
\displaystyle \frac{p^2-y^2}{p^2-2p y+y^2} \div \displaystyle \frac{\color{blue}{3 p+3 y}}{\color{red}{21p-12}} &= \displaystyle \frac{p^2-y^2}{p^2-2p y+y^2} \cdot \displaystyle \frac{\color{red}{21p-12}}{\color{blue}{3 p+3 y}} &\mbox{invert and multiply}\\
&= \displaystyle \frac{(p^2-y^2)(21p-12)}{(p^2-2py+y^2)(3p+3y)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{\color{magenta}{(p+y)(p-y)}\cdot \color{magenta}{3} \cdot (7p-4)}{\color{magenta}{(p-y)}(p-y)\cdot \color{magenta}{3} \cdot \color{magenta}{(p+y)}} &\mbox{factor numerators and denominators}\\
&= \displaystyle \frac{7p-4}{p-y} &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\
\end{array}
$$
$\displaystyle \frac{77 m^2+16 m-16}{20 m-55 m^2} \div (28 m^2 -61 m -44)$
$$
\begin{array}{cll}
\displaystyle \frac{77 m^2+16 m-16}{20 m-55 m^2} \div (28 m^2 -61 m -44) &= \displaystyle \frac{77 m^2+16 m-16}{20 m-55 m^2} \div \displaystyle \frac{\color{blue}{28 m^2 -61 m -44}}{\color{red}{1}} &\mbox{write as fraction}\\
&= \displaystyle \frac{77 m^2+16 m-16}{20 m-55 m^2} \cdot \displaystyle \frac{\color{red}{1}}{\color{blue}{28 m^2 -61 m -44}} &\mbox{invert and multiply}\\
&= \displaystyle \frac{77 m^2+16 m-16}{(20 m-55 m^2)(28 m^2 -61 m -44)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{(7m+4)(11m-4)}{5m(\color{darkorange}{4-11m})\cdot (4m-11)(7m+4)} &\mbox{factor numerators and denominators}\\
&= \displaystyle \frac{(7m+4)(11m-4)}{\color{darkorange}{-}5m(\color{darkorange}{11m-4})\cdot (4m-11)(7m+4)} &\mbox{reverse order to obtain common factor}\\
&= \displaystyle -\frac{\color{magenta}{(7m+4)}\color{magenta}{(11m-4)}}{5m\color{magenta}{(11m-4)}\cdot (4m-11)\color{magenta}{(7m+4)}} &\mbox{put negative out front and identify } \color{magenta}{\mbox{common}} \mbox{ factors}\\
&= \displaystyle -\frac{1}{5m(4m-11)} &\mbox{there's always a 1 left over if everything cancels}\\
\end{array}
$$
$\displaystyle \frac{-11 a^2+114 a+77}{11 a^2+26 a-21} \cdot \displaystyle \frac{-21 a-28}{3 a+4} \div \displaystyle \frac{11 a^2-128 a+77}{11 a-7}$
$$
\begin{array}{cll}
\displaystyle \frac{-11 a^2+114 a+77}{11 a^2+26 a-21} \cdot \displaystyle \frac{-21 a-28}{3 a+4} \div \displaystyle \frac{\color{blue}{11 a^2-128 a+77}}{\color{red}{11 a-7}} &= \displaystyle \frac{-11 a^2+114 a+77}{11 a^2+26 a-21} \cdot \displaystyle \frac{-21 a-28}{3 a+4} \cdot \displaystyle \frac{\color{red}{11 a-7}}{\color{blue}{11 a^2-128 a+77}} &\mbox{invert and multiply}\\
&= \displaystyle \frac{\color{darkorange}{(-11 a^2+114 a+77)}(-21 a-28)(11 a-7)}{(11 a^2+26 a-21)(3 a+4)(11 a^2-128 a+77)} &\mbox{multiply numerators and denominators}\\
&= \displaystyle \frac{\color{darkorange}{-(11 a^2-114 a-77)}(-21 a-28)(11 a-7)}{(11 a^2+26 a-21)(3 a+4)(11 a^2-128 a+77)} &\mbox{factor out negative}\\
&= \displaystyle \frac{-(11a +7)\color{magenta}{(a-11)}(-7)\color{magenta}{(3a+4)}\color{magenta}{(11a-7)}}{\color{magenta}{(11a-7)}(a+3)\color{magenta}{(3a+4)}(11a-7)\color{magenta}{(a-11)}} &\mbox{factor numerators and denominators}\\
&= \displaystyle \frac{-(11a +7)(-7)}{(a+3)(11a-7)} &\mbox{cancel }\color{magenta}{\mbox{common}} \mbox{ factors}\\
&= \displaystyle \frac{7(11a+7)}{(a+3)(11a-7)} &\mbox{simplify} \\
\end{array}
$$