Answer :
Therefore, the required polynomial function is f(x) = -3x³ + 5x² - 2x + 12.
Given f(-2) = 22 and f(2) = 46.Let's assume that the function f(x) is a polynomial function of degree 3. Let us take the general form of the polynomial as follows:f(x) = ax³ + bx² + cx + d Where a, b, c, and d are constants that we need to determine.Finding the value of a:To find the value of a, we can use the given information f(-2) = 22 and f(2) = 46.Substituting x = -2 in the general form of the polynomial, we get
:f(-2) = a(-2)³ + b(-2)² + c(-2) + d= -8a + 4b - 2c + d = 22
Substituting x = 2 in the general form of the polynomial, we get
:f(2) = a(2)³ + b(2)² + c(2) + d= 8a + 4b + 2c + d = 46
Solving these two equations, we get the value of a as follows:
-8a + 4b - 2c + d = 22 ... (1)8a + 4b + 2c + d = 46 ... (2)
Adding equation (1) and equation (2), we get:
0a + 8b + 0c + 2d = 68
Simplifying, we get
:4b + d = 34 ... (3)
Subtracting equation (2) from equation (1), we get
:-16a + 0b - 4c + 0d = -24
Simplifying, we get:
4a + c = 6 ... (4)
Solving equations (3) and (4), we get
:b = 5, c = -2, d = 12
.Substituting these values in the general form of the polynomial, we get:
f(x) = ax³ + bx² + cx + d= -3x³ + 5x² - 2x + 12
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