End Behavior (since leading coefficient is positive, 0.3):
Even degree ($8$): Both ends go up ($x \to \pm\infty, p(x) \to +\infty$)
Odd degree ($9$): Left end down, right end up.
But since the polynomial is degree 8, both ends go up!
3. Behavior at Each Root 🌈
Triple root ($x = 0$): The graph flattens and crosses the x-axis.
Double root ($x = 2$): The graph just touches the x-axis and turns around (bounces).
Single roots ($x = 1, -1, 3$): The graph crosses the x-axis sharply.
4. Putting It All Together: Matching the Figure 🖼️
Check for all roots at correct places.
Check that at $x = 0$, the graph flattens as it passes through.
At $x = 2$, the graph touches and bounces (does not cross).
At $x = 1, -1, 3$, the graph crosses the axis.
Both ends of the graph rise up (since degree is even and leading coefficient is positive).
If a figure shows all these features, that is the correct graph for $p(x)$!
🎯 Final Answer
Figure 1 represents the polynomial $ p(x) $!
It has all the correct roots, the right behavior at each root, and the correct end behavior.
How Roots Are Found in the Solution 🌱
Let’s see how the roots (also called zeros or x-intercepts) of the polynomial are determined step by step!
1. Start with the Polynomial
Given:
$$
p(x) = 0.3x^3(x^2 - 1)(x - 2)^2(x - 3)
$$
2. Set the Polynomial Equal to Zero
To find the roots, set $ p(x) = 0 $:
$$
0.3x^3(x^2 - 1)(x - 2)^2(x - 3) = 0
$$
A product is zero if any factor is zero!
3. Solve Each Factor Separately
$ x^3 = 0 $
Solution: $ x = 0 $
Multiplicity: 3 (because of the exponent 3)
$ x^2 - 1 = 0 $
Factor further: $ (x - 1)(x + 1) = 0 $
Solutions: $ x = 1 $, $ x = -1 $
Each has multiplicity 1
$ (x - 2)^2 = 0 $
Solution: $ x = 2 $
Multiplicity: 2 (because of the exponent 2)
$ x - 3 = 0 $
Solution: $ x = 3 $
Multiplicity: 1
4. Summary Table
Factor
Root
Multiplicity
Explanation
$ x^3 $
0
3
$ x^3 = 0 $
$ x^2 - 1 $
1, -1
1 each
$ x^2 - 1 = 0 $
$ (x - 2)^2 $
2
2
$ (x - 2)^2 = 0 $
$ x - 3 $
3
1
$ x - 3 = 0 $
5. What Does Multiplicity Mean?
Multiplicity tells how many times a root repeats.
Odd multiplicity: The graph crosses the x-axis.
Even multiplicity: The graph touches and bounces off the x-axis.
In summary:
To find the roots, set the polynomial equal to zero and solve each factor for $ x $. The exponents tell you the multiplicity (how many times each root occurs)!
