Question

question
find the graph of the polynomial given below.
$f(x)=4(x-2)(x-1)(x+8)$
select the correct answer below:

Explanation:

Step1: Find x-intercepts

Set $f(x)=0$, solve $4(x-2)(x-1)(x+8)=0$.
Roots are $x=-8, x=1, x=2$.

Step2: Determine leading term behavior

Expand leading part: $4x \cdot x \cdot x = 4x^3$. As $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to-\infty$.

Step3: Test interval signs

• For $x < -8$ (e.g., $x=-9$): $4(-11)(-10)(-1)=-440 < 0$
• For $-8 < x < 1$ (e.g., $x=0$): $4(-2)(-1)(8)=64 > 0$
• For $1 < x < 2$ (e.g., $x=1.5$): $4(-0.5)(0.5)(9.5)=-9.5 < 0$
• For $x > 2$ (e.g., $x=3$): $4(1)(2)(11)=88 > 0$

Step4: Identify y-intercept

Set $x=0$: $f(0)=4(-2)(-1)(8)=64$

Answer:

The graph will:

1. Cross the x-axis at $x=-8$, $x=1$, and $x=2$
2. Trend downward as $x\to-\infty$ and upward as $x\to+\infty$
3. Be negative when $x < -8$, positive when $-8 < x < 1$, negative when $1 < x < 2$, and positive when $x > 2$
4. Pass through the y-axis at $(0, 64)$

Match these characteristics to the provided graph options to select the correct one.