Question

question 1 of 24 step 1 of 1 construct a polynomial function with the stated properties. reduce all fractions to lowest terms. third - degree, with zeros of - 3, - 1, and 4, and a y - intercept of - 7. answer 2 points keypad keyboard shortcuts p(x) = \frac{7}{12}x^3 - \frac{9}{12}x - 7|

Explanation:

Step1: Recall Polynomial Form

A third - degree polynomial with zeros \(r_1, r_2, r_3\) can be written in factored form as \(p(x)=a(x - r_1)(x - r_2)(x - r_3)\), where \(a\) is a constant. Given zeros are \(-3\), \(-1\), and \(4\), so \(r_1=-3\), \(r_2 = - 1\), \(r_3=4\). Then the factored form is \(p(x)=a(x + 3)(x + 1)(x - 4)\).

Step2: Find the Constant \(a\)

We know that the \(y\) - intercept is the value of \(p(x)\) when \(x = 0\). The \(y\) - intercept is \(-7\), so substitute \(x = 0\) and \(p(0)=-7\) into the equation:

$$
LATEXBLOCK0
$$

Solve for \(a\): \(a=\frac{-7}{-12}=\frac{7}{12}\)

Step3: Expand the Polynomial

Substitute \(a = \frac{7}{12}\) into the factored form:

$$
LATEXBLOCK1
$$

Answer:

\(p(x)=\frac{7}{12}x^{3}-\frac{91}{12}x - 7\)