Rational Expressions
Grade 10 · Math · Study Guide
Simplifying, multiplying, dividing, adding, and subtracting algebraic fractions.
A rational expression is a fraction where the numerator and denominator are polynomials — essentially an algebraic fraction. Examples include (x² - 4)/(x + 3) or (2x)/(x² - 1). The key restriction: the denominator can never equal zero, so you must always identify values that make the denominator zero and exclude them from the domain.
Simplifying rational expressions works just like simplifying numeric fractions: factor both the numerator and denominator fully, then cancel any common factors. For instance, (x² - 4)/(x - 2) = (x+2)(x-2)/(x-2) = x + 2, provided x ≠ 2. Note: you can only cancel factors (multiplicative), never terms (additive). Canceling (x + 2) from (x + 2 + 3)/(x + 2) is a common mistake — that numerator is NOT a product of (x + 2).
Multiplying rational expressions: multiply numerators together, multiply denominators together, then simplify. Dividing: multiply by the reciprocal of the divisor, then simplify. These operations mirror what you do with numeric fractions.
Adding and subtracting rational expressions requires a common denominator, just like with regular fractions. If the denominators are different, find the least common denominator (LCD) by factoring each denominator and taking the highest power of each factor. Rewrite each fraction with the LCD, then add or subtract the numerators, keeping the denominator the same.
Key terms
- Rational Expression
- A fraction with polynomial numerator and denominator; denominator cannot equal zero.
- Domain Restriction
- Values excluded from the domain because they make the denominator equal zero.
- Simplifying
- Canceling common factors from numerator and denominator after fully factoring both.
- Least Common Denominator (LCD)
- The smallest expression divisible by all denominators in a sum or difference.
- Reciprocal
- A fraction flipped upside down; used when dividing rational expressions.
Flashcards
- Rational expression
- A fraction where numerator and denominator are polynomials
- Domain restriction
- Values that make the denominator zero — excluded from the domain
- Simplifying rational expressions
- Factor fully, then cancel common factors (not terms)
- LCD
- Least common denominator — needed to add or subtract rational expressions
- Dividing rational expressions
- Multiply the first expression by the reciprocal of the second
- Common mistake with rationals
- Canceling terms (additive) instead of factors (multiplicative) — e.g., (x+3)/3 ≠ x
Practice questions
What value must be excluded from the domain of (x + 3)/(x - 5)?
- x = -3
- x = 5
- x = -5
- x = 3
Show answer
Answer: x = 5
Set denominator to zero: x - 5 = 0 → x = 5. This value is excluded from the domain.
Simplified: (x² - 9)/(x + 3) = x - ___, where x ≠ -3.
Show answer
Answer: 3
(x²-9)=(x+3)(x-3). Cancel (x+3): result is (x-3). So the blank is 3.
To divide rational expressions, you multiply by the reciprocal of the divisor.
Show answer
Answer: True
Division of fractions: a/b ÷ c/d = a/b × d/c. The same rule applies to rational expressions.
To add (2/x) + (3/x²), the LCD is:
- x
- x²
- 2x²
- 6x²
Show answer
Answer: x²
The LCD must be divisible by both x and x². The smallest such expression is x².
You can cancel the x terms in (x + 5)/x to get 5.
Show answer
Answer: False
(x+5)/x cannot be simplified to 5. You can only cancel factors, not terms. (x+5)/x = 1 + 5/x.
Fun facts
- Rational expressions appear in physics when calculating the combined resistance of resistors in parallel circuits — the formula 1/R_total = 1/R₁ + 1/R₂ is addition of rational expressions.
- The word 'rational' shares its root with 'ratio' — a rational number (or expression) is literally one that can be written as a ratio (fraction).
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