### Home > PC3 > Chapter 8 > Lesson 8.3.5 > Problem8-156

8-156.

Let $f(x)=x^2\left(x-a\right)^2\left(x-b\right)\left(x-c\right)^3\left(x-d\right)$. Assume $a < b < 0 < c < d$.

1. Sketch a possible graph for $y = f(x)$.

Begin by sketching the axes. Mark locations on the $x$-axis for $a$, $b$, $c$, and $d$. Then sketch a polynomial curve with roots at $x = 0$, $b$, and $d$, a double root at $x = a$, and a triple root at $x = c$.

2. Solve $f(x) ≤ 0$.

Where is your curve below or touching the $x$-axis?

3. Solve $\frac { f ( x ) } { x }≤ 0$.

When do $f(x)$ and $x$ have opposite signs?
For example, where is the curve above the $x$-axis on the left side of the $x$-axis?