If _ x 1 x^3 + ax^2 + bx + c x^2 - 2x + 1 = 2026 then the value of a - c is...

Step-by-step solution

Let f(x) = x^3 + ax^2 + bx + c . For the limit to exist and be finite, the denominator (x-1)^2 must be a factor of the numerator f(x) . Let f(x) = (x-1)^2(x-k) . Expanding this, we get f(x) = (x^2-2x+1)(x-k) = x^3 - (k+2)x^2 + (2k+1)x - k . Comparing the coefficients, we have a = -(k+2) and c = -k . The limit is given by _ x 1 (x-1)^2(x-k) (x-1)^2 = _ x 1 (x-k) = 1-k . Given that the limit is 2026 , we have 1-k = 2026 k = -2025 . We need to find the value of a-c . a-c = -(k+2) - (-k) = -k - 2 + k = -2 . Answer: -2