๐ Topic Summary
The zeros of a polynomial are the $x$-values that make the polynomial equal to zero. These zeros correspond to the $x$-intercepts of the polynomial's graph. The multiplicity of a zero refers to the number of times a particular factor appears in the factored form of the polynomial. For example, in the polynomial $(x-2)^3(x+1)$, the zero $x=2$ has a multiplicity of 3, and the zero $x=-1$ has a multiplicity of 1. The multiplicity affects the behavior of the graph at the $x$-intercept; an even multiplicity means the graph touches the $x$-axis and turns around, while an odd multiplicity means the graph crosses the $x$-axis.
Understanding zeros, multiplicity, and $x$-intercepts is crucial for sketching polynomial graphs and solving polynomial equations. This quiz will test your knowledge of these fundamental concepts.
๐ค Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Zero of a polynomial |
A. The number of times a factor appears in the factored form. |
| 2. X-intercept |
B. The point where the graph crosses the x-axis. |
| 3. Multiplicity |
C. The value of $x$ that makes the polynomial equal to zero. |
| 4. Factor Theorem |
D. A theorem stating that if $f(c)=0$, then $(x-c)$ is a factor of $f(x)$. |
| 5. Polynomial Function |
E. A function that involves only non-negative integer powers of $x$. |
Matching Answers:
- ๐ข 1 - C
- ๐ 2 - B
- โ 3 - A
- ๐ก 4 - D
- โ๏ธ 5 - E
โ๏ธ Part B: Fill in the Blanks
Fill in the blanks with the correct terms.
The __________ of a polynomial is the value of $x$ that makes the polynomial equal to zero. The __________ of a zero tells us how many times the corresponding factor appears in the factored form of the polynomial. If a zero has an even multiplicity, the graph __________ the $x$-axis at that point. If a zero has an odd multiplicity, the graph __________ the $x$-axis at that point. The $x$-intercepts are also known as the __________ of the function.
Fill-in-the-Blanks Answers:
- ๐ Zero
- โ Multiplicity
- โ Touches
- โ Crosses
- ๐ Roots
๐ค Part C: Critical Thinking
Explain how the multiplicity of a zero affects the graph of a polynomial function at the x-intercept. Provide an example to illustrate your explanation.
Answer:
- ๐ก The multiplicity of a zero determines the behavior of the graph near the x-intercept.
- ๐ If the multiplicity is even, the graph touches the x-axis and turns around (it's tangent to the x-axis).
- ๐ If the multiplicity is odd, the graph crosses the x-axis.
- ๐ For example, consider $f(x) = (x-2)^2(x+1)$. The zero $x=2$ has multiplicity 2, so the graph touches the x-axis at $x=2$. The zero $x=-1$ has multiplicity 1, so the graph crosses the x-axis at $x=-1$.
