How do you combine (3n + 8)/(n^2 + 6n + 8) - ( 4n + 2)/(n^2 + n - 12)3n+8n2+6n+8−4n+2n2+n−12?

1 Answer
Mr. Mike
Apr 17, 2018

(3n+8)/(n^2+6n+8)-(4n-2)/(n^2+n-12)=(n^2+7n+20)/((n+4)(n+2)(3-n))3n+8n2+6n+8−4n−2n2+n−12=n2+7n+20(n+4)(n+2)(3−n)

Explanation:

(3n+8)/(n^2+6n+8)-(4n-2)/(n^2+n-12)3n+8n2+6n+8−4n−2n2+n−12

First factor the denominators.

(3n+8)/((n+4)(n+2))-(4n-2)/((n+4)(n-3))3n+8(n+4)(n+2)−4n−2(n+4)(n−3)

The common denominator is (n+4)(n+2)(n-3)(n+4)(n+2)(n−3)

Multiply each quotient by the appropriate factor to display the common denominator

((3n+8))/((n+4)(n+2))((n-3))/((n-3))-((4n-2))/((n+4)(n-3))((n+2))/((n+2))(3n+8)(n+4)(n+2)(n−3)(n−3)−(4n−2)(n+4)(n−3)(n+2)(n+2)

Expand the numerators using the distributive property (or FOIL if you like).

(3n^2-n-24)/((n+4)(n+2)(n-3))-(4n^2+6n-4)/((n+4)(n+2)(n-3))3n2−n−24(n+4)(n+2)(n−3)−4n2+6n−4(n+4)(n+2)(n−3)

We can combine the quotients because they have a common denominator.

(3n^2-n-24-4n^2-6n+4)/((n+4)(n+2)(n-3))3n2−n−24−4n2−6n+4(n+4)(n+2)(n−3)

Combine like terms.

(-n^2-7n-20)/((n+4)(n+2)(n-3))=(n^2+7n+20)/((n+4)(n+2)(3-n))−n2−7n−20(n+4)(n+2)(n−3)=n2+7n+20(n+4)(n+2)(3−n)

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