The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 14 (Intermediate Algebra) Rational Expressions

Overview

A rational expression is a fraction whose numerator and denominator are polynomials. Everything you already know about ordinary fractions — reducing, multiplying, dividing, and adding/subtracting — carries over unchanged; the only new wrinkle is that you must factor the polynomials first instead of factoring whole numbers. This lecture works through all four operations, plus two ideas that come before them: identifying excluded values (inputs that make the expression undefined) and evaluating an expression at a given number. The big recurring theme is that you cancel common factors top and bottom, and to do that you have to factor completely first.

Key Concepts

Excluded values. Division by zero is undefined, so a rational expression is undefined wherever its denominator equals zero. The excluded values are exactly the solutions of (denominator) $= 0$ . The numerator being zero is fine: $0$ divided by any nonzero number is just $0$ .

To find them, set the denominator equal to zero, factor, and solve. For $3x^2 + 5x = x(3x+5)$ , the factors give $x = 0$ and $x = -\tfrac{5}{3}$ , so those two values are excluded.

Evaluating an expression. Substitute the given number for every $x$ and simplify. If the substitution makes the denominator zero, the value is undefined; otherwise you get a number.

Reducing (simplifying). Factor the numerator and denominator completely, then cancel every factor they share.

Cancelling a factor against itself leaves a $1$ , never a $0$ — e.g. $\tfrac{3}{6} = \tfrac{1\cdot 3}{2\cdot 3} = \tfrac{1}{2}$ , not $\tfrac{0}{2}$ . You can always think of a lone factor as $1 \times$ (that factor).
Watch for special factoring patterns. A difference of squares factors as $a^2 - b^2 = (a-b)(a+b).$
For monomials, cancel matching variable powers: $\dfrac{15x^4y^2}{25x^2y^6} = \dfrac{3x^2}{5y^4}$ .

Multiplying. Multiply numerators together and denominators together. Factor first and cancel any common factor from any numerator against any denominator (cross-cancelling) before multiplying, so you work with small numbers.

Dividing. Multiply by the reciprocal of the divisor (flip the second fraction), then proceed exactly as in multiplication.

Least common denominator (LCD). The LCD of two denominators is their least common multiple — the smallest expression both divide into evenly. Factor each denominator; the LCD includes each distinct factor raised to the highest power in which it appears.

Numbers: $\operatorname{lcm}(8,16)=16$ (the smaller "thing that both go into," not the greatest common factor).
With variables, just take enough of each variable for both: $\operatorname{lcm}(4x^2y^5,\ 6x^4y^3z^6) = 12x^4y^5z^6$ .

Building up a fraction. To rewrite a fraction with a larger (target) denominator, multiply top and bottom by whatever factor the denominator is missing. Multiplying by $\tfrac{k}{k}=1$ never changes the value.

Adding and subtracting. Give both fractions the LCD, combine the numerators over that common denominator, then simplify (factor and cancel if possible).

When subtracting, distribute the minus sign to every term of the second numerator — the single most common mistake. For example $-(15x)$ in $\;6x-12-(15x-?)\;$ flips signs throughout.

Worked Example

Reduce $\dfrac{x^2-25}{x^2+8x+15}$ .

Factor the numerator as a difference of squares, $a^2-b^2=(a-b)(a+b)$ , and factor the trinomial denominator by finding two numbers that multiply to $15$ and add to $8$ (namely $3$ and $5$ ); then cancel the shared factor:

\begin{aligned} \frac{x^2-25}{x^2+8x+15} &= \frac{x^2-5^2}{x^2+8x+15} \\ &= \frac{(x-5)(x+5)}{(x+5)(x+3)} \\ &= \frac{x-5}{x+3}. \end{aligned}

(Excluded values: $x \neq -3$ and $x \neq -5$ , since those make the original denominator zero.)

Summary

A rational expression is a polynomial over a polynomial; treat it like a fraction, but factor before you do anything.
Excluded values are the inputs that make the denominator zero — find them by setting the denominator to $0$ and solving.
Reduce by factoring top and bottom and cancelling common factors; cancelling leaves a $1$ , not a $0$ .
Multiply straight across (factor and cross-cancel first); divide by multiplying by the reciprocal.
Recognize key patterns: difference of squares $a^2-b^2=(a-b)(a+b)$ and trinomials $x^2+bx+c=(x+m)(x+n)$ where $mn=c,\ m+n=b$ .
To add or subtract, build each fraction up to the LCD (least common multiple of the denominators), combine numerators — distributing the minus sign when subtracting — then simplify.