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Pre-Calculus Polynomial and rational functions

Operations on rational expressions (Precalc-level)

20 practice questions 0 video lessons Theory + worked examples
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Theory

A rational expression is a fraction of polynomials. You combine them with the same rules as numerical fractions — but the key move is always to factor first.

  • Multiply: factor, cancel common factors, multiply across.
  • Divide: multiply by the reciprocal of the second expression, then simplify.
  • Add / subtract: rewrite over the least common denominator (LCD), then combine numerators.
  • Complex fraction: simplify the top and bottom separately, then divide.
Track the domain. Any value that made an original denominator zero stays excluded, even if the factor cancels.
Multiplying rational expressions by factoring and canceling To multiply rational expressions, factor everything, then cancel a common factor that appears in a numerator and a denominator before multiplying across. (x+2)(x−1)x−1×xx+2= xcancel (x−1) and (x+2)
Multiplying: factor, cancel \((x-1)\) and \((x+2)\), then multiply the survivors.
Adding rational expressions with a common denominator To add fractions with unlike denominators, rewrite each over the least common denominator, then add the numerators. 1x+1x+1=(x+1)+xx(x+1)=2x+1x(x+1)LCD = x(x+1)
Adding: rewrite over the LCD \(x(x+1)\), then add the numerators.

The four operations on \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\):

\[\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd},\qquad \dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}\]
\[\dfrac{a}{b}\pm\dfrac{c}{d}=\dfrac{ad\pm bc}{bd}\ \text{(or over the LCD)}\]
multiply numerators and denominators; divide by multiplying by the reciprocal; add over a common denominator
Cancel factors, not terms. You may cancel a whole factor like \((x-1)\), never a single term inside a sum.

How to combine rational expressions

  1. Factor every numerator and denominator completely.
  2. For \(\times/\div\): turn division into multiplication by the reciprocal, then cancel common factors.
  3. For \(+/-\): find the LCD, rewrite each fraction over it, and combine numerators.
  4. Simplify the result and state the excluded values.
Example 1 — Multiply and simplify
Simplify \(\dfrac{x^2+x-2}{x-1}\cdot\dfrac{x}{x+2}\).
Solution

Factor everything, then cancel common factors before multiplying.

\(=\)\(\dfrac{(x+2)(x-1)}{x-1}\cdot\dfrac{x}{x+2}\)
\(=\)\(\dfrac{\cancel{(x+2)}\,\cancel{(x-1)}}{\cancel{(x-1)}}\cdot\dfrac{x}{\cancel{(x+2)}}\)
\(=\)\(x\)

(with \(x\neq 1,-2\)).

product simplifies to x, with x not equal to 1 or negative 2
Example 2 — Divide by multiplying by the reciprocal
Simplify \(\dfrac{x^2-9}{x}\div\dfrac{x+3}{x^2}\).
Solution

Dividing is multiplying by the reciprocal of the second fraction.

\(=\)\(\dfrac{x^2-9}{x}\cdot\dfrac{x^2}{x+3}\)
\(=\)\(\dfrac{(x-3)(x+3)}{x}\cdot\dfrac{x^2}{x+3}\)
\(=\)\((x-3)\cdot x\)
\(=\)\(x(x-3)\)
quotient simplifies to x times x minus 3
Example 3 — Add unlike denominators
Simplify \(\dfrac{1}{x}+\dfrac{1}{x+1}\).
Solution

The least common denominator is \(x(x+1)\). Rewrite each fraction over it, then add numerators.

\(=\)\(\dfrac{(x+1)}{x(x+1)}+\dfrac{x}{x(x+1)}\)
\(=\)\(\dfrac{(x+1)+x}{x(x+1)}\)
\(=\)\(\dfrac{2x+1}{x(x+1)}\)
sum is 2x plus 1 over x times x plus 1
Example 4 — A complex fraction
Simplify \(\dfrac{\ \dfrac{1}{x}-\dfrac{1}{2}\ }{x-2}\).
Solution

Combine the top over its LCD \(2x\) first, then divide.

\(\text{top}\)\(=\)\(\dfrac{2-x}{2x}\)
\(=\)\(\dfrac{2-x}{2x}\cdot\dfrac{1}{x-2}\)
\(=\)\(\dfrac{-(x-2)}{2x(x-2)}\)
\(=\)\(-\dfrac{1}{2x}\)
complex fraction simplifies to negative 1 over 2x

Common pitfalls

Cancel factors, not terms. In \(\dfrac{x+2}{x}\) you cannot cancel the \(x\)’s — the top is a sum, not a product.
Flip the correct fraction when dividing. Multiply by the reciprocal of the divisor (the second one) only.
Keep excluded values. A value that made an original denominator zero remains excluded even after a factor cancels.

Frequently asked questions

How do you multiply rational expressions?

Factor every numerator and denominator, cancel any factor common to a top and a bottom, then multiply the remaining factors across.

How do you divide rational expressions?

Multiply the first expression by the reciprocal of the second, then factor and cancel as with multiplication.

How do you add or subtract rational expressions?

Rewrite each over the least common denominator, combine the numerators, then simplify.

Can you cancel terms across a fraction?

No — only whole factors. You may cancel \((x-1)\) from top and bottom, but not a lone \(x\) from \(\dfrac{x+2}{x}\).

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