Lesson 88: Multiplying and Dividing Rational Expressions

New Concept

If $a$, $b$, $c$, and $d$ are nonzero polynomials, then $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.

What’s next

Next, you’ll apply these rules to multiply, divide, and simplify expressions involving polynomials, just as you would with regular fractions.

Property

If $a$, $b$, $c$, and $d$ are nonzero polynomials, then $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.

Explanation

Multiplying these is like regular fractions! Just multiply straight across: top times top, and bottom times bottom. After you multiply, simplify the result by canceling out any common factors. This gives you the cleanest possible answer and makes you look like a math wizard, ready for any challenge!

Examples

$\frac{2x^3}{5y^2} \cdot \frac{15x}{8y^3} = \frac{30x^4}{40y^5} = \frac{3x^4}{4y^5}$
$\frac{x-2}{x+3} \cdot \frac{x^2-9}{x^2-4} = \frac{x-2}{x+3} \cdot \frac{(x-3)(x+3)}{(x-2)(x+2)} = \frac{x-3}{x+2}$
$\frac{4x^2}{3y} \cdot \frac{9y^3}{2x} = \frac{36x^2y^3}{6xy} = 6xy^2$

See how factoring first makes a tricky multiplication problem surprisingly simple. This is a core part of multiplying rational expressions.

Example Problem

Multiply $\frac{8}{2x-12} \cdot (x^2 - 3x - 18)$ and simplify.

Property

If $a$, $b$, $c$, and $d$ are nonzero polynomials, then $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$.

Explanation

Division is just a sneaky multiplication problem! Remember the golden rule: "Keep, Change, Flip." Keep the first fraction, change division to multiplication, and flip the second fraction upside down. From there, you just multiply like normal and simplify your final answer. Easy peasy, you've totally got this!

Examples

$\frac{4x^3}{y^2} \div \frac{2x}{y^3} = \frac{4x^3}{y^2} \cdot \frac{y^3}{2x} = 2x^2y$
$\frac{x^2-16}{x+2} \div (x-4) = \frac{(x-4)(x+4)}{x+2} \cdot \frac{1}{x-4} = \frac{x+4}{x+2}$
$\frac{a^2+2a+1}{a^2} \div \frac{a+1}{a} = \frac{(a+1)^2}{a^2} \cdot \frac{a}{a+1} = \frac{a+1}{a}$

Remember that division is just multiplication in disguise. Let's use that idea to tackle this problem on dividing rational expressions.

Example Problem

Find the quotient: $\frac{x^2+6x+8}{x^3} \div \frac{x+4}{x}$.

Property

A strategic method to simplify expressions is to factor all polynomials in the numerators and denominators before multiplying. This approach reveals common factors that can be canceled out early, streamlining the entire multiplication process and making it much more manageable.

Explanation

Why wrestle with huge, messy polynomials when you can shrink them first? Factoring before you multiply is the ultimate math shortcut. It lets you spot and cancel out matching pieces right away, which makes the whole problem easier and helps you avoid mistakes. It’s like cleaning your room before a party!

Examples

$\frac{x^2-4}{x^2+2x} \cdot \frac{x}{x-2} = \frac{(x-2)(x+2)}{x(x+2)} \cdot \frac{x}{x-2} = 1$
$\frac{5}{2x-12} \cdot (x^2-36) = \frac{5}{2(x-6)} \cdot \frac{(x-6)(x+6)}{1} = \frac{5(x+6)}{2}$
$\frac{2x^2n+4xn}{2x} \cdot \frac{10}{10xn+20n} = \frac{2xn(x+2)}{2x} \cdot \frac{10}{10n(x+2)} = 1$