Question

Given that 2x³-5x² - 4x+8 = (Ax - 1)(x-B)(x + 1) + C for all values of x, find the

values of A, B and C.

Answer 1

A=2, B=3, C=5

Explanation:

Two polynomials are equal for all values of the variable if the corresponding coefficients are the same. The way to solve the problem is to multiply the RHS, rewrite it in a more orderly way

`(Ax-1)(x-B)(x+1)+C = \\Ax^3+(A-AB-1)x^2+(B-AB-1)x +B+C`

Now, for the two polynomials to be the same we need to have, at the same time

`x^3: A=2\\x^2: A-AB-1= -5\\x:B-AB-1=-4\\1: B+C=8`

You might notice that we have more conditions than variables, but you can consider one of the two in the middle as a "check" option.

Now, grabbing the value from the first condition into the second we get B=3. Replacing both value in the third we see that the equality still holds, and replacing into the fourth we get C =5.