Warm-up
Three quick questions from earlier lessons.
Discover
Simplify 6/9.
- 2/3
- 6/9 cannot be simplified
Key idea
Rational expressions work like fractions.
Factor numerators and denominators, then cancel common factors (never terms). Multiply numerators and denominators; to divide, multiply by the reciprocal. Keep the values that make an original denominator zero excluded.
Worked example
Simplify (x² − 9)/(x² + 3x).
(x − 3)(x + 3) / (x(x + 3))
Factor.
(x − 3)/x, for x ≠ 0 and x ≠ −3
Cancel x + 3.
Your turn, guided
Your turn: the value of (x² − 4)/(x − 2), simplified, at x = 3.
x + 2 at x = 3.
Practice ★
- Simplify (x² − 4) ÷ (x − 2), that is, the fraction with numerator x² − 4 and denominator x − 2. Which answer is right?
- x + 2, for x ≠ 2
- x − 2
- x + 2, for every x
- Simplify (x² − 25) ÷ (x − 5), that is, the fraction with numerator x² − 25 and denominator x − 5. Which answer is right?
- x + 5, for x ≠ 5
- x − 5
- x + 5, for every x
- Multiply and simplify: (x + 8)/(x − 6) · (x − 6)/(x + 9).
Watch out!
Practice ★★
- Divide and simplify: x/(x + 3) ÷ x²/(x + 3).
- Divide and simplify: x/(x + 6) ÷ x²/(x + 6).
- Multiply and simplify: (x + 8)/(x − 6) · (x − 6)/(x + 9).
Exit ticket
Three questions to check you have got it.
- Simplify (x² − 4) ÷ (x − 2), that is, the fraction with numerator x² − 4 and denominator x − 2. Which answer is right?
- x + 2, for x ≠ 2
- x − 2
- x + 2, for every x
- Simplify (x² − 25) ÷ (x − 5), that is, the fraction with numerator x² − 25 and denominator x − 5. Which answer is right?
- x + 5, for x ≠ 5
- x − 5
- x + 5, for every x
- Simplify (x² − 9) ÷ (x − 3), that is, the fraction with numerator x² − 9 and denominator x − 3. Which answer is right?
- x + 3, for x ≠ 3
- x − 3
- x + 3, for every x
Summary card
Multiplying and dividing rational expressions
Factor, cancel common factors, multiply; divide by multiplying by the reciprocal.
Watch out: cancel factors, never terms.
End of the lesson
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