In Algebra 2, understanding the multiplicity of zeros is key to analyzing polynomial functions. A zero of a polynomial is a value of $x$ that makes the polynomial equal to zero. The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. For example, in the polynomial $(x-2)^3(x+1)$, the zero $x=2$ has a multiplicity of 3, while the zero $x=-1$ has a multiplicity of 1. The multiplicity affects how the graph behaves at that $x$-intercept: odd multiplicities result in the graph crossing the x-axis, while even multiplicities cause the graph to 'bounce' or touch the x-axis and turn around.
Understanding the relationship between the factored form of a polynomial, its zeros, their multiplicities, and the resulting graph is crucial for sketching and analyzing polynomial functions.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Zero | A. The number of times a factor appears in a polynomial. |
| 2. Multiplicity | B. The point where the graph intersects or touches the x-axis. |
| 3. Root | C. The value of x that makes the polynomial equal to zero. |
| 4. x-intercept | D. Another name for a zero of a polynomial. |
| 5. Factor | E. An expression that divides evenly into a polynomial. |
The __________ of a zero determines how the graph behaves at the x-intercept. If the multiplicity is __________, the graph crosses the x-axis. If the multiplicity is __________, the graph touches the x-axis and turns around. The __________ form of a polynomial reveals its zeros and their multiplicities. The end behavior of the graph is determined by the polynomial's __________ term.
Explain how you can determine the end behavior of a polynomial function given its equation, and how this relates to the leading coefficient and degree of the polynomial.
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