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 \(-4\), \(-3\), and \(5\), and a \(y\)-intercept of \(-9\).

Answer

\(p(x) =\)

Solution

Step 1: Since the polynomial is third-degree with zeros at -4, -3, and 5, it can be expressed as \[ p(x) = a(x + 4)(x + 3)(x - 5) \] Step 2: Use the y-intercept to find \( a \). When \( x = 0 \), \( p(0) = -9 \). \[ -9 = a(0 + 4)(0 + 3)(0 - 5) \] \[ -9 = a(4)(3)(-5) \] \[ -9 = -60a \] \[ a = \frac{9}{60} = \frac{3}{20} \] Step 3: Substitute \( a \) back into the polynomial \[ p(x) = \frac{3}{20}(x + 4)(x + 3)(x - 5) \]