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
Step 1: Understand the Factor Theorem. The Factor Theorem states that if (x-c) is a factor of a polynomial P(x), then P(c) = 0. Step 2: Identify the polynomial and the factor. The given polynomial is P(x) = 2x^3+4x^2+px+7. The given factor is (x+2). Comparing (x+2) with (x-c), we find that c = -2. Step 3: Apply the Factor Theorem. Since (x+2) is a factor of P(x), we must have P(-2) = 0. Substitute x = -2 into the polynomial: P(-2) = 2(-2)^3 + 4(-2)^2 + p(-2) + 7 Step 4: Simplify the expression and solve for p. 2(-8) + 4(4) - 2p + 7 = 0 -16 + 16 - 2p + 7 = 0 0 - 2p + 7 = 0 -2p + 7 = 0 -2p = -7 p = (-7)/(-2) p = (7)/(2) The value of p is (7)/(2).