Question: Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms…

Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms.

Third-degree, with zeros of \(-4\), \(-2\), and \(3\), and a \(y\)-intercept of \(-14\).

\(p(x) =\)

Solution

Since the polynomial \( p(x) \) is a third-degree polynomial with zeros at -4, -2, and 3, it can be expressed as: \[ p(x) = a(x + 4)(x + 2)(x - 3) \] To find the value of \( a \), use the y-intercept \( p(0) = -14 \): \[ p(0) = a(0 + 4)(0 + 2)(0 - 3) = a \cdot 4 \cdot 2 \cdot (-3) = -24a \] Set this equal to -14: \[ -24a = -14 \] Solve for \( a \): \[ a = \frac{14}{24} = \frac{7}{12} \] Substitute \( a \) back into the polynomial: \[ p(x) = \frac{7}{12}(x + 4)(x + 2)(x - 3) \]