Question

PLEASE HELP 100 POINTS !! What two rational expressions sum to 4x+2/x^2−9x+8?

Answer 1

4x+2 34/7 -6/7

--------- = ------------ +--------

( x-8) (x-1) ( x-8) (x-1)

Explanation:

4x+2

---------

x^2−9x+8

Factor the denominator

x^2 -9x +8 = ( x-8) (x-1)

4x+2 A B

--------- = ------------ +--------

( x-8) (x-1) ( x-8) (x-1)

Multiply each side by ( x-8) (x-1)

4x+2 = A ( x-1) + B ( x-8)

4x+2 = Ax -A + Bx - B8

Making two equations

4x = AX + Bx

Factoring out the x

4 = A + B

2 = -A - 8B

Adding these equation to eliminate A

4 = A + B

2 = -A - 8B

--------------------

6 = -7B

-6/7 = B

Now finding A

4 = A -6/7

4 + 6/7 = A

28/7 + 6/7 = A

34/7 = A

4x+2 34/7 -6/7

--------- = ------------ +--------

( x-8) (x-1) ( x-8) (x-1)

Answer 2

4x+2/x^2−9x+8

~Factor denominator

4x+2/(x-1)(x-8)

~Create partial fraction using the denominator

a(x-1)(x-8)/x-1 + b(x-1)(x-8)/x-8

~Simplify

4x + 2 = a(x - 8) + b(x - 1)

4x + 2 = xa + xb - 8a - b

4x + 2 = x(a + b) − 8a - B

a + b = 4 and -8a - b = 2

You solve the system and get a = -6/7 and b = 34/7

~Denominator root 1:

a = -6/7

~Denominator root 8:

b = 34/7

~Plug the solutions

-6/7(x-1) + 34/7(x-8)

Best of Luck!