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) =

Explanation:

Step1: Identify the roots

The polynomial has roots at $x = - 2$, $x = 4$. Since the graph touches the $x$-axis at $x = 4$, the root $x = 4$ has an even - multiplicity (let's assume multiplicity 2 for simplicity). The root $x=-2$ has multiplicity 1. So the polynomial has factors $(x + 2)$ and $(x - 4)^2$.

Step2: Determine the leading - coefficient

Let the polynomial be $y(x)=a(x + 2)(x - 4)^2$. We can use the $y$-intercept. When $x = 0$, $y(0)=a(0 + 2)(0 - 4)^2$. From the graph, when $x = 0$, $y=-2$. So $-2=a(2)(16)$, which gives $a=-\frac{1}{16}$.

Step3: Write the polynomial equation

$y(x)=-\frac{1}{16}(x + 2)(x - 4)^2=-\frac{1}{16}(x + 2)(x^{2}-8x + 16)=-\frac{1}{16}(x^{3}-8x^{2}+16x+2x^{2}-16x + 32)=-\frac{1}{16}(x^{3}-6x^{2}+32)=-\frac{1}{16}x^{3}+\frac{3}{8}x^{2}-2$

Answer:

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