+4622 Find A and B such that 5x+2/x^2-7x-30=A/x-10+B/x+3. Write your answer in the form (A,B)
You should bracket the terms that are ambiguous, such as:2/x^2-7x. Is it: 2 / (x^2) - 7? or 2/(x^2 - 7)?....etc., so that there is no question as to what you want.
+129899 Tertre....I believe this is a partial fraction decomposition.....I think we must have.....
( 5x + 2) / (x^2 - 7x - 30) = A / (x - 10) + B / ( x + 3)
( 5x + 2) / (x^2 - 7x - 30) = [ A(x + 3) + B(x - 10) ] / [ (x - 10) (x +3) ]
Expanding the denominator on the right will produce the denominator on the left so we can ignore these and solve this
5x + 2 = A(x +3) + B(x - 10)
5x + 2 = (A + B)x + [ 3A - 10B]
Equate coefficients and we have the following system :
A + B = 5 → B = 5 - A (1)
3A - 10B = 2 (2)
Sub (1) into (2)
3A - 10 (5 - A) = 2
3A - 50 + 10A = 2
13A = 52 → A = 4
And B = 5 - A = 5 - 4 = 1
Check for yourself that
4/ ( x - 10) + 1/ ( x + 3) produces the rational function [5x + 2] / [ x^2 - 7x - 30 ]
P.S. - The use of parentheses/brackets would make it easier to figure out what you're asking.......
