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 \(-5\), \(-1\), and \(3\), and a \(y\)-intercept of \(-10\).

Answer: \(2\) Points

\(p(x) =\)

Solution

To construct the polynomial function \( p(x) \) with the given properties: First, since the polynomial is of third degree with zeros at -5, -1, and 3, it can be expressed as: \[ p(x) = a(x + 5)(x + 1)(x - 3) \] Next, use the y-intercept to find the value of \( a \). The y-intercept is \( p(0) = -10 \). \[ p(0) = a(0 + 5)(0 + 1)(0 - 3) = a(5)(1)(-3) = -15a \] \[ -15a = -10 \\ a = \frac{2}{3} & \] Therefore, the polynomial function is: \[ p(x) = \frac{2}{3}(x + 5)(x + 1)(x - 3) \]