Question

write an equation for the polynomial graphed below
question help: video message instructor

Explanation:

Step1: Identify the roots

The polynomial has roots at \(x=-3\), \(x = 1\), and \(x = 4\). So the polynomial has factors \((x + 3)\), \((x - 1)\), and \((x - 4)\). The general form of the polynomial is \(y(x)=a(x + 3)(x - 1)(x - 4)\).

Step2: Find the value of \(a\)

The graph passes through the point \((0,4)\). Substitute \(x = 0\) and \(y=4\) into \(y(x)=a(x + 3)(x - 1)(x - 4)\):
\[

$$\begin{align*} 4&=a(0 + 3)(0 - 1)(0 - 4)\\ 4&=a(3)(-1)(-4)\\ 4&=12a\\ a&=\frac{1}{3} \end{align*}$$

\]

Answer:

\(y(x)=\frac{1}{3}(x + 3)(x - 1)(x - 4)\)