The function f is defined by f(x) = 2x³ + 3x² + cx + 8, where c is a constant. In the xy-plane, the graph of f intersects the x-axis at the three points (−4, 0), (1/2, 0), and (p, 0). What is the value of c?

Answers

Answer 1

The Value of c is 18.

Given function is$f(x) = 2x^3 + 3x^2 + cx + 8$Given points on the x-axis are (-4, 0), (1/2, 0), and (p, 0)

For the given function to intersect the x-axis at (-4,0), one of the factors should be (x+4)For the given function to intersect the x-axis at (1/2, 0), one of the factors should be (2x-1)For the given function to intersect the x-axis at (p, 0), one of the factors should be (x-p).

Therefore, the function can be written as,$f(x) = a(x+4)(2x-1)(x-p)$We know that, when x=0, y intercept = f(0)Since it passes through the point (8,0),Substituting x=0 in the above expression,$f(0) = a(0+4)(2(0)-1)(0-p)$$\implies 8ap = 8 \implies ap = 1$.

Now, let's expand the function$f(x) = a(x+4)(2x-1)(x-p)$$f(x) = 2ax^3 + ax^2 - 9ax - 4a = 2x^3 + 3x^2 + cx + 8$Comparing the coefficients,2a = 23a = -c-9a = 0-4a = 8Solving these equations, we get,a = -2, c = 18.

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