The function f is defined by where c is a constant. In the xy-plane, the graph of f intersects the x-axis at the three points (-4,0), and What is the value of c?
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Choice A is correct. The given zeros can be used to set up an equation to solve for c. Substituting –4 for x and 0 for y yields or Alternatively, since –4, and p are zeros of the polynomial function it follows that Were this polynomial multiplied out, the constant term would be (We can see this without performing the full expansion.) Since it is given that this value is 8, it follows that or rather, Substituting 2 for p in the polynomial function yields and after multiplying the factors one finds that the coefficient of the x term, or the value of c, is –18.
Choice B is not the correct answer. This value is a misunderstood version of the value of p, not c, and the relationship between the zero and the factor (if is the zero of a polynomial, its corresponding factor is ) has been confused.
Choice C is not the correct answer. This is the value of p, not c. Using this value as the third factor of the polynomial will reveal that the value of c is -18.
Choice D is not the correct answer. This represents a sign error in the final step in determining the value of c.
Students must understand the zeros of a polynomial function and how they are used to construct algebraic representations of polynomials.