Question

question #2 - 20 points
the zeros of a polynomial function are: -2, 0, 1, 3

1. use the zeros to write a polynomial in factored form. (4 points)
2. determine the degree of the polynomial. (2 points)
3. describe the end behavior of the polynomial function. (4 points)
4. determine the y-intercept of the polynomial (2 points)
5. sketch a graph of the polynomial. (8 points)

Explanation:

Step1: Write factored form from zeros

For a zero $r$, the factor is $(x-r)$. Using zeros $-2,0,1,3$:
$P(x) = a(x+2)(x-0)(x-1)(x-3) = ax(x+2)(x-1)(x-3)$
(where $a$ is a non-zero real constant; we use $a=1$ for the simplest polynomial)

Step2: Identify polynomial degree

Count the number of linear factors: there are 4 distinct linear factors, so the degree equals the number of factors.

Step3: Analyze end behavior

For a degree 4 (even) polynomial with positive leading coefficient ($a=1>0$):
As $x\to+\infty$, $P(x)\to+\infty$; as $x\to-\infty$, $P(x)\to+\infty$

Step4: Calculate y-intercept

Set $x=0$ in $P(x)=x(x+2)(x-1)(x-3)$:
$P(0) = 0(0+2)(0-1)(0-3) = 0$

Step5: Prepare for graph sketching

1. Plot zeros: $x=-2, 0, 1, 3$ (x-intercepts)
2. Plot y-intercept: $(0,0)$
3. Test values between zeros:

• $x=-1$: $P(-1)=(-1)(1)(-2)(-4)=-8$
• $x=0.5$: $P(0.5)=(0.5)(2.5)(-0.5)(-2.5)=1.5625$
• $x=2$: $P(2)=(2)(4)(1)(-1)=-8$

1. Apply end behavior: both ends rise to $+\infty$

Answer:

1. $P(x) = ax(x+2)(x-1)(x-3)$ (or the simplest form $P(x)=x(x+2)(x-1)(x-3)$ when $a=1$)
2. 4
3. As $x\to+\infty$, $P(x)\to+\infty$; as $x\to-\infty$, $P(x)\to+\infty$
4. $(0, 0)$ (or $y=0$)
5. A graph with:

• x-intercepts at $(-2,0)$, $(0,0)$, $(1,0)$, $(3,0)$
• y-intercept at $(0,0)$
• Points $(-1,-8)$, $(0.5, 1.5625)$, $(2,-8)$ plotted
• Ends rising upward as $x$ approaches $+\infty$ and $-\infty$, crossing the x-axis at each zero.