This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Step 1: Apply the Remainder Theorem. The Remainder Theorem states that if a polynomial P(x) is divided by (ax-b), the remainder is P((b)/(a)). Given P(x) = 4x^3 + kx^2 + 7x - 23 and the divisor is 2x-5. Here, a=2 and b=5, so we need to evaluate P((5)/(2)). The remainder is given as 7, so P((5)/(2)) = 7. Step 2: Substitute x = (5)/(2) into the polynomial and set it equal to the remainder. 4((5)/(2))^3 + k((5)/(2))^2 + 7((5)/(2)) - 23 = 7 Step 3: Simplify the terms. 4((125)/(8)) + k((25)/(4)) + (35)/(2) - 23 = 7 (125)/(2) + (25k)/(4) + (35)/(2) - 23 = 7 Step 4: Combine the constant terms. (125)/(2) + (35)/(2) = (160)/(2) = 80 So the equation becomes: 80 + (25k)/(4) - 23 = 7 57 + (25k)/(4) = 7 Step 5: Isolate the term with k and solve for k. (25k)/(4) = 7 - 57 (25k)/(4) = -50 25k = -50 × 4 25k = -200 k = (-200)/(25) k = -8 The value of the constant k is -8. What's next? 📸