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: Identify the given information for the cubic polynomial. Let the zeroes of the cubic polynomial be , , . The general form of a cubic polynomial is P(x) = ax^3 + bx^2 + cx + d. The relationships between the zeroes and coefficients are: Sum of zeroes: + + = -(b)/(a) Sum of the product of zeroes taken two at a time: + + = (c)/(a) Product of zeroes: = -(d)/(a) From the problem statement: Sum of zeroes = -2 Sum of the product of its zeroes taken two at a time = 7 Product of its zeroes = 16 Step 2: Set up the equations based on the given values. We can assume a=1 to find a cubic polynomial. So, we have: -(b)/(1) = -2 b = 2 (c)/(1) = 7 c = 7 -(d)/(1) = 16 d = -16 Step 3: Form the cubic polynomial. Substitute the values of a, b, c, d into the general form P(x) = ax^3 + bx^2 + cx + d. P(x) = 1x^3 + 2x^2 + 7x - 16 P(x) = x^3 + 2x^2 + 7x - 16 The cubic polynomial is x^3 + 2x^2 + 7x - 16.