How do you add \frac { a - 5} { a ^ { 2} + 1} + \frac { a + 1} { a ^ { 2} - 1}a−5a2+1+a+1a2−1?

1 Answer
Andy Y.
Nov 22, 2017

(2a^2-6a+6)/((a^2+1)(a-1))2a2−6a+6(a2+1)(a−1)

Explanation:

You add fractions by ensuring the denominators are the same. One way to get the same denominator is to multiply each fraction by the other denominator, like so

(a-5)/(a^2+1) xx (color(red)(a^2-1))/(color(red)(a^2-1))+(a+1)/(a^2-1)xx(color(blue)(a^2+1))/(color(blue)(a^2+1))a−5a2+1×a2−1a2−1+a+1a2−1×a2+1a2+1

With identical denominators, we can now add the numerators

((a-5)(a^2-1)+(a+1)(a^2+1))/((a^2+1)(a^2-1))(a−5)(a2−1)+(a+1)(a2+1)(a2+1)(a2−1)

It may be useful to factor out (a^2-1)(a2−1), giving

((a-5)(a-1)(a+1)+(a+1)(a^2+1))/((a^2+1)(a-1)(a+1))(a−5)(a−1)(a+1)+(a+1)(a2+1)(a2+1)(a−1)(a+1)

This allows us to cancel the (a+1)(a+1) terms

((a-5)(a-1)cancel((a+1))+cancel((a+1))(a^2+1))/((a^2+1)(a-1)cancel((a+1)))

Expanding the numerator gives

(a^2-6a+5+a^2+1)/((a^2+1)(a-1))

Combing like terms in the numerator

(2a^2-6a+6)/((a^2+1)(a-1))

Related questions
Impact of this question
1037 views around the world
You can reuse this answer
Creative Commons License