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.

Respuesta :

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

[tex]g(x)=k(x-a)(x-b)(x-c),[/tex]

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

Therefore:

[tex]p(x)=k(x+3)(x+2)(x-1).[/tex]

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

[tex]p(0)=-14=k(0+3)(0+2)(0-1).[/tex]

Solving for k, we get:

[tex]\begin{gathered} -14=-6k, \\ k=-\frac{14}{-6}, \\ k=\frac{14}{6}, \\ k=\frac{7}{3}. \end{gathered}[/tex]

Finally:

[tex]p(x)=\frac{7}{3}(x+3)(x+2)(x-1).[/tex]

Answer:

[tex]p(x)=\frac{7}{3}(x+3)(x+2)(x-1).[/tex]

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