H Algebra 2/Trig · Unit 4
Simplify a fraction that has fractions in the numerator, the denominator, or both.
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One denominator
Find the LCD of the small fractions.
Multiply the numerator and the denominator of the big fraction by that LCD.
A power of x
If a small denominator is x², the LCD is x².
Multiplying by x² clears every small denominator.
Binomial denominators
Simplify the top and the bottom as their own additions or subtractions.
Then multiply by the reciprocal.
Excluded values
x = 0 is excluded when a small fraction has x in the denominator.
Also exclude any value that makes a later denominator 0.
The small LCD is x
The LCD of the small fractions is x.
Multiply the numerator and the denominator by x.
x + 2x + 3
Factor.
(x + 2)(x + 3)
(x + 2)(x + 3)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −3 and x ≠ 0
Answer: (x + 2)(x + 3), x ≠ −3 and x ≠ 0
The LCD of the small fractions is x.
Multiply the numerator and the denominator by x.
3x + 6x + 2
Factor.
3(x + 2)(x + 2)
Cancel the common factor.
3
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −2 and x ≠ 0
Answer: 3, x ≠ −2 and x ≠ 0
The LCD of the small fractions is x.
Multiply the numerator and the denominator by x.
5x + 10x + 2
Factor.
5(x + 2)(x + 2)
Cancel the common factor.
5
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −2 and x ≠ 0
Answer: 5, x ≠ −2 and x ≠ 0
The LCD of the small fractions is x.
Multiply the numerator and the denominator by x.
4x + 82x + 4
Factor.
4(x + 2)2(x + 2)
Cancel the common factor.
2
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −2 and x ≠ 0
Answer: 2, x ≠ −2 and x ≠ 0
The LCD of the small fractions is x.
Multiply the numerator and the denominator by x.
2x + 4x + 4
Factor.
2(x + 2)(x + 4)
2(x + 2)(x + 4)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −4 and x ≠ 0
Answer: 2(x + 2)(x + 4), x ≠ −4 and x ≠ 0
The LCD of the small fractions is x.
Multiply the numerator and the denominator by x.
6x + 32x + 1
Factor.
3(2x + 1)(2x + 1)
Cancel the common factor.
3
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −12 and x ≠ 0
Answer: 3, x ≠ −12 and x ≠ 0
The small LCD is x²
The LCD of the small fractions is x².
Multiply the numerator and the denominator by x².
x² + 2xx² + 3x + 2
Factor.
x(x + 2)(x + 1)(x + 2)
Cancel the common factor.
x(x + 1)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −2, x ≠ −1, and x ≠ 0
Answer: x(x + 1), x ≠ −2, x ≠ −1, and x ≠ 0
The LCD of the small fractions is x².
Multiply the numerator and the denominator by x².
x² + 3xx² + 5x + 6
Factor.
x(x + 3)(x + 2)(x + 3)
Cancel the common factor.
x(x + 2)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −3, x ≠ −2, and x ≠ 0
Answer: x(x + 2), x ≠ −3, x ≠ −2, and x ≠ 0
The LCD of the small fractions is x².
Multiply the numerator and the denominator by x².
x² + 4xx² + 7x + 12
Factor.
x(x + 4)(x + 3)(x + 4)
Cancel the common factor.
x(x + 3)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −4, x ≠ −3, and x ≠ 0
Answer: x(x + 3), x ≠ −4, x ≠ −3, and x ≠ 0
The LCD of the small fractions is x².
Multiply the numerator and the denominator by x².
2x² + 6xx² + 4x + 3
Factor.
2x(x + 3)(x + 1)(x + 3)
Cancel the common factor.
2x(x + 1)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −3, x ≠ −1, and x ≠ 0
Answer: 2x(x + 1), x ≠ −3, x ≠ −1, and x ≠ 0
The LCD of the small fractions is x².
Multiply the numerator and the denominator by x².
3x² + 3xx² + 2x + 1
Factor.
3x(x + 1)(x + 1)²
Cancel the common factor.
3x(x + 1)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −1 and x ≠ 0
Answer: 3x(x + 1), x ≠ −1 and x ≠ 0
The LCD of the small fractions is x².
Multiply the numerator and the denominator by x².
4x² + 4xx² + 3x + 2
Factor.
4x(x + 1)(x + 1)(x + 2)
Cancel the common factor.
4x(x + 2)
Exclude any value that makes a denominator 0, including x = 0.
x ≠ −2, x ≠ −1, and x ≠ 0
Answer: 4x(x + 2), x ≠ −2, x ≠ −1, and x ≠ 0
Binomial denominators
Simplify the numerator and the denominator first.
Numerator:
x² − x − 1x(x + 1)
Denominator:
x² + x + 1x(x + 1)
Dividing by a fraction means multiply by its reciprocal.
Cancel the common factor.
x² − x − 1x² + x + 1
Exclude every value that makes a denominator 0.
x ≠ −1 and x ≠ 0
Answer: x² − x − 1x² + x + 1, x ≠ −1 and x ≠ 0
Simplify the numerator and the denominator first.
Numerator:
6x + 9x(x + 3)
Denominator:
2x + 3x(x + 3)
Dividing by a fraction means multiply by its reciprocal.
3x(2x + 3)(x + 3)x(x + 3)(2x + 3)
Cancel the common factor.
3
Exclude every value that makes a denominator 0.
x ≠ −3 and x ≠ 0
Answer: 3, x ≠ −3 and x ≠ 0
Simplify the numerator and the denominator first.
Numerator:
x + 5(x − 3)(x + 1)
Denominator:
5x − 11(x − 3)(x + 1)
Dividing by a fraction means multiply by its reciprocal.
(x + 5)(x − 3)(x + 1)(x − 3)(x + 1)(5x − 11)
Cancel the common factor.
(x + 5)(5x − 11)
Exclude every value that makes a denominator 0.
x ≠ −1 and x ≠ 3
Answer: (x + 5)(5x − 11), x ≠ −1 and x ≠ 3
Simplify the numerator and the denominator first.
Numerator:
2x − 5(x + 2)(x − 1)
Denominator:
2x + 1(x + 2)(x − 1)
Dividing by a fraction means multiply by its reciprocal.
(2x − 5)(x + 2)(x − 1)(x + 2)(x − 1)(2x + 1)
Cancel the common factor.
(2x − 5)(2x + 1)
Exclude every value that makes a denominator 0.
x ≠ −2 and x ≠ 1
Answer: (2x − 5)(2x + 1), x ≠ −2 and x ≠ 1
Simplify the numerator and the denominator first.
Numerator:
2x − 16(x + 4)(x − 2)
Denominator:
2x + 2(x + 4)(x − 2)
Dividing by a fraction means multiply by its reciprocal.
2(x − 8)(x + 4)(x − 2)2(x + 4)(x − 2)(x + 1)
Cancel the common factor.
(x − 8)(x + 1)
Exclude every value that makes a denominator 0.
x ≠ −4 and x ≠ 2
Answer: (x − 8)(x + 1), x ≠ −4 and x ≠ 2
Simplify the numerator and the denominator first.
Numerator:
x² − x + 4(x − 2)(x + 1)
Denominator:
4x − 5(x − 2)(x + 1)
Dividing by a fraction means multiply by its reciprocal.
Cancel the common factor.
x² − x + 4(4x − 5)
Exclude every value that makes a denominator 0.
x ≠ −1 and x ≠ 2
Answer: x² − x + 4(4x − 5), x ≠ −1 and x ≠ 2
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