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Try It 7.61: Simplifying 3x2+7x+104x+2+1x+5\frac{\frac{3}{x^2+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}} Using the LCD

To simplify the complex rational expression 3x2+7x+104x+2+1x+5\frac{\frac{3}{x^2+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}} using the least common denominator (LCD) method:

Step 1. Find the LCD of all inner fractions. The inner denominators are x2+7x+10x^2+7x+10, (x+2)(x+2), and (x+5)(x+5). Factor the trinomial: x2+7x+10=(x+2)(x+5)x^2+7x+10 = (x+2)(x+5). Since this factored form already contains both of the other denominators, the LCD is (x+2)(x+5)(x+2)(x+5).

Step 2. Multiply the numerator and denominator by the LCD, (x+2)(x+5)(x+2)(x+5). (x+2)(x+5)⋅3(x+2)(x+5)(x+2)(x+5)⋅4x+2+(x+2)(x+5)⋅1x+5\frac{(x+2)(x+5) \cdot \frac{3}{(x+2)(x+5)}}{(x+2)(x+5) \cdot \frac{4}{x+2} + (x+2)(x+5) \cdot \frac{1}{x+5}}

Step 3. Simplify by canceling common factors in each term. 34(x+5)+1(x+2)\frac{3}{4(x+5) + 1(x+2)}

Step 4. Distribute and combine like terms in the denominator. 34x+20+x+2=35x+22\frac{3}{4x + 20 + x + 2} = \frac{3}{5x + 22}

The simplified expression is 35x+22\frac{3}{5x + 22}.

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Updated 2026-04-30

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References

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OpenStax

Intermediate Algebra @ OpenStax

Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

Algebra

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