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# Rational Expressions of Polynomials

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Find all A and B such that $$\frac{4x}{x^2-8x+15} = \frac{A}{x-3} + \frac{B}{x-5}$$ for all x besides 3 and 5. Express your answer as an ordered pair in the form (A, B).

Not quite sure how to solve this. Thanks in advance.

Mar 30, 2018

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4x                                  A                   B

__________     =          ______   +       _____

x^2 - 8x + 15                  x  - 3               x - 5

This is known as a partial-fraction decomposition

Note that the denominator on the left factors as  ( x - 3)  (x  - 5)

Multiply both sides by this and we have

4x   =  A(x - 5)   +  B ( x - 3)     simplify

4x  = Ax - 5A + Bx - 3B

4x + 0 =  (A + B)x  + (- 5A -3B)

Equating coefficients, we have this system

A + B  = 4         ⇒  B  = 4 - A      (1)

-5A  - 3B  =  0   ⇒  5A  + 3B  =  0     (2)

Sub (1)  into (2)   and we have that

5A + 3(4 - A)  = 0

5A + 12 - 3A  = 0

2A  =  -12

A  = -6

And  B  =  4 - A   =   4 - (-6)   =  10

Verify  that

-6/ ( x - 3)  +  10/ (x  - 5)     =   the left hand side of the original equation

Mar 30, 2018