Question

Find the value of A, B and C, given that 2z^2 + z + C = A (z+1)^2 + B (z+1) + 4 for all values of z. Please show the step by step process. :)

Answer 1

2z² + z + C = A (z + 1)² + B (z + 1) + 4

Expand the right side completely and collect terms with the same power of z :

2z² + z + C = A (z² + 2z + 1) + B (z + 1) + 4

2z² + z + C = Az² + 2Az + A + Bz + B + 4

2z² + z + C = Az² + (2A + B)z + A + B + 4

Now match up terms on either side that have the same power of z. This gives

2 = A ................. (2nd power)

1 = 2A + B ......... (1st power)

C = A + B + 4 .... (0th power)

Solve for A, B, and C. The first equation gives A = 2 right away, so in the second equation we get

1 = 2*2 + B

B = -3

and in the third equation we get

C = 2 + (-3) + 4

C = 3