choosing the equation of a polynomial function
which of the following polynomial functions could represent the graph?
$2x^3 + 2x^2 - 4x$
$3x^3 + 3x^2 - 6x$
$x^3 + x^2 - 2x$
$-3x^3 - 3x^2 + 6x$

Question

Accepted Answer

# Answer:
$3x^{3}+3x^{2}-6x$
# Explanation:
## Step1: Analyze the roots of the polynomial
From the graph, the x - intercepts are $x=- 2$, $x = 0$ and $x=1$. So the polynomial can be written in factored form as $y=a(x + 2)x(x - 1)$, where $a$ is a leading coefficient. Expand this:
$$
\begin{align*}
y&=a(x + 2)x(x - 1)\
&=a(x^{2}+2x)x(x - 1)\
&=a(x^{3}+2x^{2})(x - 1)\
&=a(x^{4}+2x^{3}-x^{3}-2x^{2})\
&=a(x^{4}+x^{3}-2x^{2})
\end{align*}
$$
Wait, actually, the degree of the polynomial from the graph (it has two turning points, so degree is 3) is 3. So the correct factored form is $y=a(x + 2)x(x - 1)=a(x^{3}+x^{2}-2x)$

## Step2: Use the point $(-1,6)$ to find $a$
Substitute $x=-1$ and $y = 6$ into $y=a(x^{3}+x^{2}-2x)$:
$$
\begin{align*}
6&=a((-1)^{3}+(-1)^{2}-2\times(-1))\
6&=a(-1 + 1+2)\
6&=a\times2\
a&=3
\end{align*}
$$

## Step3: Write the polynomial
Substitute $a = 3$ into $y=a(x^{3}+x^{2}-2x)$, we get $y=3x^{3}+3x^{2}-6x$

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze the roots of the polynomial

Step2: Use the point \((-1,6)\) to find \(a\)

Step3: Write the polynomial

Answer: