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# Multiplication of Rational Expressions

## Multiply and reduce fractions with variables in the denominator

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Multiplication of Rational Expressions

### Multiplication of Rational Expressions

The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers, so let’s start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately:

abcd=acbd\begin{align*}\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d}\end{align*}

#### Multiplying Rational Expressions Involving Monomials

1. Multiply the following: a16b84b35a2\begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}\end{align*}.

Cancel common factors from the numerator and denominator. The common factors are 4, a\begin{align*}a\end{align*}, and b3\begin{align*}b^3\end{align*}. Canceling them out leaves 14b515a=120ab5\begin{align*}\frac{1}{4b^5} \cdot \frac{1}{5a} = \frac{1}{20ab^5}\end{align*}.

2. Multiply 9x24y221x4\begin{align*}9x^2 \cdot \frac{4y^2}{21x^4}\end{align*}.

Rewrite the problem as a product of two fractions: 9x214y221x4\begin{align*}\frac{9x^2}{1} \cdot \frac{4y^2}{21x^4}\end{align*} Then cancel common factors from the numerator and denominator.

The common factors are 3 and x2\begin{align*}x^2\end{align*}. Canceling them out leaves 314y27x2=12y27x2\begin{align*}\frac{3}{1} \cdot \frac{4y^2}{7x^2} = \frac{12y^2}{7x^2}\end{align*}.

#### Multiplying Rational Expressions Involving Polynomials

When multiplying rational expressions involving polynomials, first we need to factor all polynomial expressions as much as we can. Then we follow the same procedure as before.

Multiply 4x+123x2xx29\begin{align*}\frac{4x+12}{3x^2} \cdot \frac{x}{x^2-9}\end{align*}.

Factor all polynomial expressions as much as possible: 4(x+3)3x2x(x+3)(x3)\begin{align*}\frac{4(x+3)}{3x^2} \cdot \frac{x}{(x+3)(x-3)}\end{align*}

The common factors are x\begin{align*}x\end{align*} and (x+3)\begin{align*}(x + 3)\end{align*}. Canceling them leaves 43x1(x3)=43x(x3)=43x29x\begin{align*}\frac{4}{3x} \cdot \frac{1}{(x-3)} = \frac{4}{3x(x-3)} = \frac{4}{3x^2-9x}\end{align*}.

#### Multiplying a Rational Expression by a Polynomial

When we multiply a rational expression by a whole number or a polynomial, we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous examples.

Multiply 3x+184x2+19x5(x2+3x10)\begin{align*}\frac{3x+18}{4x^2+19x-5} \cdot (x^2+3x-10)\end{align*}.

Rewrite the expression as a product of fractions: 3x+184x2+19x5x2+3x101\begin{align*}\frac{3x+18}{4x^2+19x-5} \cdot \frac{x^2+3x-10}{1}\end{align*}

Factor polynomials: 3(x+6)(x+5)(4x1)(x2)(x+5)1\begin{align*}\frac{3(x+6)}{(x+5)(4x-1)} \cdot \frac{(x-2)(x+5)}{1}\end{align*}

The common factor is (x+5)\begin{align*}(x + 5)\end{align*}. Canceling it leaves 3(x+6)(4x1)(x2)1=3(x+6)(x2)(4x1)=3x2+12x364x1\begin{align*}\frac{3(x+6)}{(4x-1)} \cdot \frac{(x-2)}{1} = \frac{3(x+6)(x-2)}{(4x-1)} = \frac{3x^2+12x-36}{4x-1}\end{align*}

### Example

#### Example 1

Multiply 12x2x6x21x2+7x+64x227x+18\begin{align*}\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18}\end{align*}.

Factor polynomials: (3x+2)(4x3)(x+1)(x1)(x+1)(x+6)(4x3)(x6)\begin{align*}\frac{(3x+2)(4x-3)}{(x+1)(x-1)} \cdot \frac{(x+1)(x+6)}{(4x-3)(x-6)}\end{align*}.

The common factors are (x+1)\begin{align*}(x + 1)\end{align*} and (4x3)\begin{align*}(4x - 3)\end{align*}. Canceling them leaves (3x+2)(x1)(x+6)(x6)=(3x+2)(x+6)(x1)(x6)=3x2+20x+12x27x+6\begin{align*}\frac{(3x+2)}{(x-1)} \cdot \frac{(x+6)}{(x-6)} = \frac{(3x+2)(x+6)}{(x-1)(x-6)} = \frac{3x^2+20x+12}{x^2-7x+6}\end{align*}

### Review

Multiply the following rational expressions and reduce the answer to lowest terms.

1. x32y32y2x\begin{align*}\frac{x^3}{2y^3} \cdot \frac{2y^2}{x}\end{align*}
2. 2xy24y5x\begin{align*}\frac{2x}{y^2} \cdot \frac{4y}{5x}\end{align*}
3. 2xy2y2x3\begin{align*}2xy \cdot \frac{2y^2}{x^3}\end{align*}
4. 4y21y29y32y1\begin{align*}\frac{4y^2-1}{y^2-9} \cdot \frac{y-3}{2y-1}\end{align*}
5. 6aba2a3b3b2\begin{align*}\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}\end{align*}
6. 33a252011a3\begin{align*}\frac{33a^2}{-5} \cdot \frac{20}{11a^3}\end{align*}
7. 2x2+2x24x2+3xx2+x6x+4\begin{align*}\frac{2x^2+2x-24}{x^2+3x} \cdot \frac{x^2+x-6}{x+4}\end{align*}
8. xx5x28x+15x23x\begin{align*}\frac{x}{x-5} \cdot \frac{x^2-8x+15}{x^2-3x}\end{align*}
9. 5x2+16x+336x225(6x2+5x)\begin{align*}\frac{5x^2+16x+3}{36x^2-25} \cdot (6x^2+5x)\end{align*}
10. x2+7x+10x29x23x3x2+4x4\begin{align*}\frac{x^2+7x+10}{x^2-9} \cdot \frac{x^2-3x}{3x^2+4x-4}\end{align*}
11. x2+8x+167x2+9x+27x+2x2+4x\begin{align*}\frac{x^2+8x+16}{7x^2+9x+2} \cdot \frac{7x+2}{x^2+4x}\end{align*}

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