H Algebra 2/Trig · Unit 4
Divide rational expressions by multiplying by the reciprocal, cancel common factors, and keep every excluded value.
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Multiply by the reciprocal
Dividing by a fraction means multiply by its reciprocal.
Keep the first fraction, flip the second, and multiply.
Factor, then cancel
Factor every numerator and denominator before you cancel.
Cancel a factor only when it appears in a numerator and a denominator.
Excluded values
A value that makes an original denominator 0 is still excluded.
Flipping the second fraction means its old numerator is now a denominator.
Opposite binomials
2 − x and x − 2 are opposites.
Their quotient is −1. The minus sign stays in the answer.
One binomial cancels
Dividing by a fraction means multiply by its reciprocal.
x² + 3x − 4x + 3 × 2x + 4
Factor x² + 3x − 4.
(x + 4)(x − 1)
Write the product as one fraction, then cancel common factors.
2(x + 4)(x − 1)(x + 3)(x + 4)
Cancel (x + 4).
2(x − 1)(x + 3)
A value that makes an original denominator 0 is still excluded.
x ≠ −4 and x ≠ −3
Answer: 2(x − 1)(x + 3), x ≠ −4 and x ≠ −3
Dividing by a fraction means multiply by its reciprocal.
x + 5x − 2 × x + 1x + 5
Write the product as one fraction, then cancel common factors.
(x + 5)(x + 1)(x − 2)(x + 5)
Cancel (x + 5).
(x + 1)(x − 2)
A value that makes an original denominator 0 is still excluded.
x ≠ −5, x ≠ −1, and x ≠ 2
Answer: (x + 1)(x − 2), x ≠ −5, x ≠ −1, and x ≠ 2
Dividing by a fraction means multiply by its reciprocal.
2x + 1x + 4 × 32x + 1
Write the product as one fraction, then cancel common factors.
3(2x + 1)(x + 4)(2x + 1)
Cancel (2x + 1).
3(x + 4)
A value that makes an original denominator 0 is still excluded.
x ≠ −4 and x ≠ −12
Answer: 3(x + 4), x ≠ −4 and x ≠ −12
Dividing by a fraction means multiply by its reciprocal.
x − 6x + 2 × x + 22
Write the product as one fraction, then cancel common factors.
(x − 6)(x + 2)2(x + 2)
Cancel (x + 2).
(x − 6)2
A value that makes an original denominator 0 is still excluded.
x ≠ −2
Answer: (x − 6)2, x ≠ −2
Dividing by a fraction means multiply by its reciprocal.
3x − 3x + 1 × 4x − 1
Write the product as one fraction, then cancel common factors.
12(x − 1)(x + 1)(x − 1)
Cancel (x − 1).
12(x + 1)
A value that makes an original denominator 0 is still excluded.
x ≠ −1 and x ≠ 1
Answer: 12(x + 1), x ≠ −1 and x ≠ 1
Dividing by a fraction means multiply by its reciprocal.
x + 85 × x − 3x + 8
Write the product as one fraction, then cancel common factors.
(x + 8)(x − 3)5(x + 8)
Cancel (x + 8).
(x − 3)5
A value that makes an original denominator 0 is still excluded.
x ≠ −8 and x ≠ 3
Answer: (x − 3)5, x ≠ −8 and x ≠ 3
Both fractions have quadratic numerators or denominators
Dividing by a fraction means multiply by its reciprocal.
2x² + 5x − 3x² − 1 × x − 12x − 1
Factor 2x² + 5x − 3.
(2x − 1)(x + 3)
Factor x² − 1.
(x − 1)(x + 1)
Write the product as one fraction, then cancel common factors.
(2x − 1)(x + 3)(x − 1)(x − 1)(x + 1)(2x − 1)
Cancel (2x − 1) and (x − 1).
(x + 3)(x + 1)
A value that makes an original denominator 0 is still excluded.
x ≠ −1, x ≠ 12, and x ≠ 1
Answer: (x + 3)(x + 1), x ≠ −1, x ≠ 12, and x ≠ 1
Dividing by a fraction means multiply by its reciprocal.
x² + x − 122x² − 8 × x − 2x − 3
Factor x² + x − 12.
(x + 4)(x − 3)
Factor 2x² − 8.
2(x − 2)(x + 2)
Write the product as one fraction, then cancel common factors.
(x + 4)(x − 3)(x − 2)2(x − 2)(x + 2)(x − 3)
Cancel (x − 3) and (x − 2).
(x + 4)2(x + 2)
A value that makes an original denominator 0 is still excluded.
x ≠ −2, x ≠ 2, and x ≠ 3
Answer: (x + 4)2(x + 2), x ≠ −2, x ≠ 2, and x ≠ 3
Dividing by a fraction means multiply by its reciprocal.
x² − 4x² + 5x + 6 × x + 3x − 2
Factor x² − 4.
(x − 2)(x + 2)
Factor x² + 5x + 6.
(x + 2)(x + 3)
Write the product as one fraction, then cancel common factors.
(x − 2)(x + 2)(x + 3)(x + 2)(x + 3)(x − 2)
Cancel (x − 2) and (x + 2) and (x + 3).
1
A value that makes an original denominator 0 is still excluded.
x ≠ −3, x ≠ −2, and x ≠ 2
Answer: 1, x ≠ −3, x ≠ −2, and x ≠ 2
Dividing by a fraction means multiply by its reciprocal.
x² − 9x² − x − 6 × x − 3x + 3
Factor x² − 9.
(x − 3)(x + 3)
Factor x² − x − 6.
(x − 3)(x + 2)
Write the product as one fraction, then cancel common factors.
(x − 3)²(x + 3)(x − 3)(x + 2)(x + 3)
Cancel (x − 3) and (x + 3).
(x − 3)(x + 2)
A value that makes an original denominator 0 is still excluded.
x ≠ −3, x ≠ −2, and x ≠ 3
Answer: (x − 3)(x + 2), x ≠ −3, x ≠ −2, and x ≠ 3
Dividing by a fraction means multiply by its reciprocal.
2x² + 7x + 3x² − 4 × x + 22x + 1
Factor 2x² + 7x + 3.
(2x + 1)(x + 3)
Factor x² − 4.
(x − 2)(x + 2)
Write the product as one fraction, then cancel common factors.
(2x + 1)(x + 3)(x + 2)(x − 2)(x + 2)(2x + 1)
Cancel (2x + 1) and (x + 2).
(x + 3)(x − 2)
A value that makes an original denominator 0 is still excluded.
x ≠ −2, x ≠ −12, and x ≠ 2
Answer: (x + 3)(x − 2), x ≠ −2, x ≠ −12, and x ≠ 2
Dividing by a fraction means multiply by its reciprocal.
x² + 2x − 8x² − 16 × x + 4x − 2
Factor x² + 2x − 8.
(x + 4)(x − 2)
Factor x² − 16.
(x − 4)(x + 4)
Write the product as one fraction, then cancel common factors.
(x + 4)²(x − 2)(x − 4)(x + 4)(x − 2)
Cancel (x + 4) and (x − 2).
(x + 4)(x − 4)
A value that makes an original denominator 0 is still excluded.
x ≠ −4, x ≠ 2, and x ≠ 4
Answer: (x + 4)(x − 4), x ≠ −4, x ≠ 2, and x ≠ 4
An opposite binomial
Dividing by a fraction means multiply by its reciprocal.
x² − 42 − x × x − 1x + 2
Factor x² − 4.
(x − 2)(x + 2)
Factor 2 − x.
−(x − 2)
Write the product as one fraction, then cancel common factors.
(x − 2)(x + 2)(x − 1)−(x − 2)(x + 2)
Cancel (x − 2) and (x + 2).
1 − x
A value that makes an original denominator 0 is still excluded.
x ≠ −2, x ≠ 1, and x ≠ 2
Answer: 1 − x, x ≠ −2, x ≠ 1, and x ≠ 2
Dividing by a fraction means multiply by its reciprocal.
x² − 93 − x × x + 1x + 3
Factor x² − 9.
(x − 3)(x + 3)
Factor 3 − x.
−(x − 3)
Write the product as one fraction, then cancel common factors.
(x − 3)(x + 3)(x + 1)−(x − 3)(x + 3)
Cancel (x − 3) and (x + 3).
−x − 1
A value that makes an original denominator 0 is still excluded.
x ≠ −3, x ≠ −1, and x ≠ 3
Answer: −x − 1, x ≠ −3, x ≠ −1, and x ≠ 3
Dividing by a fraction means multiply by its reciprocal.
2x² − x − 11 − x × x + 42x + 1
Factor 2x² − x − 1.
(2x + 1)(x − 1)
Factor 1 − x.
−(x − 1)
Write the product as one fraction, then cancel common factors.
(2x + 1)(x − 1)(x + 4)−(x − 1)(2x + 1)
Cancel (2x + 1) and (x − 1).
−x − 4
A value that makes an original denominator 0 is still excluded.
x ≠ −4, x ≠ −12, and x ≠ 1
Answer: −x − 4, x ≠ −4, x ≠ −12, and x ≠ 1
Dividing by a fraction means multiply by its reciprocal.
x² − 6x + 82 − x × x + 3x − 4
Factor x² − 6x + 8.
(x − 2)(x − 4)
Factor 2 − x.
−(x − 2)
Write the product as one fraction, then cancel common factors.
(x − 2)(x − 4)(x + 3)−(x − 2)(x − 4)
Cancel (x − 2) and (x − 4).
−x − 3
A value that makes an original denominator 0 is still excluded.
x ≠ −3, x ≠ 2, and x ≠ 4
Answer: −x − 3, x ≠ −3, x ≠ 2, and x ≠ 4
Dividing by a fraction means multiply by its reciprocal.
x² − 11 − x × x − 5x + 1
Factor x² − 1.
(x − 1)(x + 1)
Factor 1 − x.
−(x − 1)
Write the product as one fraction, then cancel common factors.
(x − 1)(x + 1)(x − 5)−(x − 1)(x + 1)
Cancel (x − 1) and (x + 1).
5 − x
A value that makes an original denominator 0 is still excluded.
x ≠ −1, x ≠ 1, and x ≠ 5
Answer: 5 − x, x ≠ −1, x ≠ 1, and x ≠ 5
Dividing by a fraction means multiply by its reciprocal.
x² − 4x − 55 − x × x − 2x + 1
Factor x² − 4x − 5.
(x − 5)(x + 1)
Factor 5 − x.
−(x − 5)
Write the product as one fraction, then cancel common factors.
(x − 5)(x + 1)(x − 2)−(x − 5)(x + 1)
Cancel (x − 5) and (x + 1).
2 − x
A value that makes an original denominator 0 is still excluded.
x ≠ −1, x ≠ 2, and x ≠ 5
Answer: 2 − x, x ≠ −1, x ≠ 2, and x ≠ 5
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