SOLUTION: If the equality x^3 = A(x - 1)(x - 2)(x - 3) + B(x - 1)(x - 2) + C(x - 2) + D is an identity, what is A + B + C + D?

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Question 41563: If the equality x^3 = A(x - 1)(x - 2)(x - 3) + B(x - 1)(x - 2) + C(x - 2) + D is an identity, what is A + B + C + D?
Answer by kev82(148) About Me  (Show Source):
You can put this solution on YOUR website!
Hi,
Because it's an identity you can shove in any values you like for x and get some simultaneous equations to solve. However solving a 4x4 set of simultaneous equations isn't much fun. By factoring we can choose suitable values of x that let us read off the values of the coefficients.
Factoring gives

Taking x=2 gives D=8
Then rearranging to get

And this time choosing x=1 gives C=7
Rearranging again (it looks messy but it's OK, trust me!)

Now choosing x=3 gives B=6
Finally

Choose x=4 to find A=1
8+7+6+1=22, which is your answer.
Hope that helps,
Kev

Message from question poster
I accidentally messed up the end of the problem, it should have been
C(x - 1), but using your problem I was able to come up with the solution. I had decided while working the problem that I would have to factor in some way, but I had not thought of factoring in that manner. I also knew the fact that it was an identity had to come into play.
Fixing the problem I got A=1, B = 6, C = 7 and D=1.
Can you tell me if this is correct??
Yes, it is. You could of course check the answer by multiplying out all the brackets and you should find you getsomething which is always true. Like 0=0.