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! $$\frac{a}{b} \cdot \frac{c}{d}=\frac{ac}{bd}$$ Great News! The SAME RULE APPLIES to rational expressions.
Example: $\frac{2x^4}{-3} \cdot \frac{-20}{-36x^3}$
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
Multiplying Rational Expressions
Examples
$\frac{\tau^2-121 \alpha^2}{3\tau^2-34\tau \alpha+11 \alpha^2} \cdot \frac{9\tau^2-6\tau \alpha+\alpha^2}{9\tau^2- \alpha^2}$
$\frac{121 t^2-121 t s+121 s^2}{7 t s} \cdot \frac{5 t^2 s}{11 t^2-11 t s+11 s^2}$
$(3 n^2-11 n -4) \cdot \frac{n^2 -3 n-28}{3 n^2-20 n -7}$
$\frac{176 t^2 r+132 t r^2}{16 t^2-9 r^2} \cdot \frac{4 t-3 r}{44 t r}$
Dividing Rational Expressions
Recall: fraction division. $$\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}$$ Great News! The SAME RULE APPLIES to rational expressions.
Example: $\frac{\xi^2-15\xi+44}{\xi^2-8\xi+16} \div \frac{\xi^2-22\xi+121}{7 \xi^2-112}$
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
Note: steps 1,2, and 3 above are the same steps for multiplication with only the additional "invert and multiply" step.
Dividing Rational Expressions
Examples
$\frac{\phi^2-y^2}{\phi^2-2\phi y+y^2} \div \frac{3 \phi+3 y}{21\phi-12}$
$\frac{77 \mu^2+16 \mu-16}{20 \mu-55 \mu^2} \div (28 \mu^2 -61 \mu -44)$
$\frac{-11 \alpha^2+114 \alpha+77}{11 \alpha^2+26 \alpha-21} \cdot \frac{-21 \alpha-28}{3 \alpha+4} \div \frac{11 \alpha^2-128 \alpha+77}{11 \alpha-7}$
One More Important Example: Reduce the rational expression. $$\frac{x^2+2x+1}{x^2+3}$$ Indeed. Some things are oh so very tempting.
But PLEASE remember...
