Problem
Find the remainder when the polynomial \(3x^3 - 5x^2 + 2x - 7\) is divided by the binomial \(x - 2\).
Solution
Step 1 ：\(3x^3 - 5x^2 + 2x - 7 = (x - 2)Q(x) + R(x)\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder.
Step 2 ：Substitute the value of \(x = 2\) into the polynomial equation: \ \(3(2)^3 - 5(2)^2 + 2(2) - 7 = 2Q(2) + R(2)\). \ Simplify to get: \(24 - 20 + 4 - 7 = 2Q(2) + R(2)\), or \(1 = 2Q(2) + R(2)\).
Step 3 ：Since \(x - 2 = 0\) when \(x = 2\), the remainder \(R(x) = R(2) = 1 - 2Q(2)\). But the remainder must be a constant, so \(Q(2) = 0\) and \(R(2) = 1\).
