Question

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Third-degree, with zeros of -3,-2, and 1, and a y-intercept of -14.

Answer 1

Recall that the general form of a third-degree polynomial is:

$g(x)=k(x-a)(x-b)(x-c),$

where k is a constant, and a, b, and c are the zeros of the polynomial.

Therefore:

$p(x)=k(x+3)(x+2)(x-1).$

Now, to determine the value of k, we consider the y-intercept:

$p(0)=-14=k(0+3)(0+2)(0-1).$

Solving for k, we get:

$\begin{gathered} -14=-6k, \\ k=-\frac{14}{-6}, \\ k=\frac{14}{6}, \\ k=\frac{7}{3}. \end{gathered}$

Finally:

$p(x)=\frac{7}{3}(x+3)(x+2)(x-1).$

Answer:

$p(x)=\frac{7}{3}(x+3)(x+2)(x-1).$