Question

write an equation for the polynomial graphed below
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
y(x) =
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Explanation:

Step1: Identify the roots

The roots of the polynomial are the x - values where the graph crosses or touches the x - axis. Here, the roots are $x=-3,x = - 2,x=2,x = 4$. So the polynomial has factors of $(x + 3),(x + 2),(x - 2),(x - 4)$.

Step2: Determine the multiplicity

Since the graph crosses the x - axis at each root, each root has a multiplicity of 1.

Step3: Find the leading - coefficient

Let the polynomial be $y(x)=a(x + 3)(x + 2)(x - 2)(x - 4)$. We can use the y - intercept (the point $(0,3)$) to find the value of $a$. Substitute $x = 0$ and $y=3$ into the equation:
\[

$$\begin{align*} 3&=a(0 + 3)(0 + 2)(0 - 2)(0 - 4)\\ 3&=a(3)\times(2)\times(-2)\times(-4)\\ 3&=a\times48\\ a&=\frac{3}{48}=\frac{1}{16} \end{align*}$$

\]

Step4: Write the polynomial equation

\[y(x)=\frac{1}{16}(x + 3)(x + 2)(x - 2)(x - 4)\]

Answer:

$y(x)=\frac{1}{16}(x + 3)(x + 2)(x - 2)(x - 4)$