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Algebraic fractions: addition, multiplication and division

The four operations on fractions with letters instead of numbers: a common denominator for addition and subtraction, cancelling in multiplication and division, and dividing a sum of cubes by a binomial.

An algebraic fraction behaves exactly like an ordinary one - only the numerator and denominator hold expressions with letters. To add two of them you need a common denominator; to multiply, you multiply numerators and denominators, and division becomes multiplication by the reciprocal. The one new caution concerns the denominator: it must not be zero, so letters in a denominator always carry an assumption with them.

Algebraic fractions

Addition and subtraction

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

\frac{a}{b} - \frac{c}{d^{2}} + \frac{d}{a} = \frac{aad^{2} - abc + bdd^{2}}{abd^{2}} = \frac{a^{2}d^{2} - abc + bd^{3}}{abd^{2}}

\begin{aligned}\frac{2}{x + 1} + \frac{4}{x + 2} &= \frac{2(x + 2) + 4(x + 1)}{(x + 1)(x + 2)} = \frac{2x + 4 + 4x + 4}{x^{2} + 2x + x + 2} = \\&= \frac{6x + 8}{x^{2} + 3x + 2}\end{aligned}

Multiplication and division

\displaystyle \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

\displaystyle \frac{a}{b} : \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}

\frac{2a}{3b} : \frac{a^{2}b}{6} = \frac{2a}{3b} \cdot \frac{6}{a^{2}b} = \frac{12a}{3a^{2}b^{2}} = \frac{4}{ab^{2}}

\frac{2a}{3b} : \frac{a^{2}b}{6} \cdot \frac{ab}{2} = \frac{2a}{3b} \cdot \frac{6}{a^{2}b} \cdot \frac{ab}{2} = \frac{12a^{2}b}{6a^{2}b^{2}} = \frac{2}{b}

Review exercises

Multiply and simplify the results.

\text{a)}\quad (2a + 4b)(a - 3b) = 2a^{2} - 6ab + 4ab - 12b^{2} = 2a^{2} - 2ab - 12b^{2}

\text{b)}\quad (9s^{2} + 3)(s^{2} - 4) = 9s^{4} - 36s^{2} + 3s^{2} - 12 = 9s^{4} - 33s^{2} - 12

Simplify each of the following expressions to a single algebraic fraction.

\text{a)}\quad \frac{ab}{c} + \frac{cb}{a} = \frac{a^{2}b + bc^{2}}{ac}

\text{b)}\quad \frac{ab}{c} - 1 = \frac{ab}{c} - \frac{c}{c} = \frac{ab - c}{c}

\begin{aligned}\text{c)}\quad \left(\frac{ab}{c} + \frac{ac}{b}\right) + \frac{bc}{a} &= \frac{ab^{2} + ac^{2}}{bc} + \frac{bc}{a} = \\&= \frac{a^{2}b^{2} + a^{2}c^{2} + b^{2}c^{2}}{abc}\end{aligned}

Perform the division.

\frac{a^{3} + 8b^{3}}{a + 2b}

(a^{3} + 8b^{3}) : (a + 2b) = a^{2} - 2ab + 4b^{2}

\begin{array}{rrrr}a^{3} & & & + \, 8b^{3} \\a^{3} & + \, 2a^{2}b & & \\ \hline0 & - \, 2a^{2}b & & \\ & - \, 2a^{2}b & - \, 4ab^{2} & \\ \hline & 0 & + \, 4ab^{2} & + \, 8b^{3} \\ & & 4ab^{2} & + \, 8b^{3} \\ \hline & & & 0\end{array}

(a^{2} - 2ab + 4b^{2})(a + 2b) = a^{3} + 8b^{3}

Frequently asked questions

How do you add two algebraic fractions?

Bring them to a common denominator by multiplying each fraction by the missing factor. For a/b + c/d the common denominator is bd, and the sum is (ad + bc)/bd.

How do you divide one fraction by another?

Multiply by the reciprocal of the second one: a/b : c/d equals a/b · d/c, that is ad/bc. After multiplying it is worth cancelling any common factors.

What do you do with a whole number in an expression with a fraction?

Write it as a fraction with the same denominator. In ab/c − 1 the one becomes c/c, and the whole expression is (ab − c)/c.

What is a³ + 8b³ divided by a + 2b?

a² − 2ab + 4b². This is the familiar sum of cubes: a³ + 8b³ is a³ + (2b)³, and such a sum always divides by a + 2b exactly.

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Multiplication and division of algebraic expressions

Factoring algebraic expressions