1. b) Algebraically, how can we recognize that r is a root of
1. a) multiplicity k?
1. x − r is a factor k times;
i.e., (x − r)k is a factor.
For example, in this polynomial,
(x − 3)4(x + 1)5
3 is a root of multiplicity 4, and −1 is a root of multiplicity 5.
3. What is a point of inflection?
A point where a graph changes concavity; from concave upward to concave downward, or vice-versa.
In each of these graphs, the origin is a point of inflection. At the origin, each graph changes from concave downward to concave upward.
As the calculus student will learn, at a point of inflection the second derivative is 0.
4. a) What happens to the graph at a root of
even multiplicity?
4. The graph touches the x-axis (is tangent to it), but does
4. not cross it.
4. b) What happens to the graph at a root of odd multiplicity?
4. The graph crosses the x-axis, and it is a point of inflection.
Problem 1. Name all the roots, and sketch the graph, of y = x³.
The roots are 0, 0, 0. Here is the graph.
Problem 2. Name all the roots, and sketch the graph, of y = x4.
The roots are 0, 0, 0, 0. Here is the graph.
Example. Name the roots of this polynomial. What is its degree? Sketch its graph.
y = (x + 2)²(x − 1)³
Answer. −2 is a root of multiplicity 2, and 1 is a root of multiplicity 3. These are the 5 roots:
−2, −2, 1, 1, 1.
This polynomial is of the 5th degree, which is odd. Therefore, the graph begins on the left below the x-axis.
−2 is a root of even multiplicity -- at −2 the graph is tangent to the x-axis. 1 is a root of odd multiplicity. 1 is a point of inflection.
Problem 3. Name the roots of this polynomial. What is its degree? Sketch its graph.
y = x³(x + 2)4(x − 3)5
This polynomial is of degree 3 + 4 + 5 = 12. These are the 12 roots:
0, 0, 0, −2, −2, −2, −2, 3, 3, 3, 3, 3.
Here is the graph.
This polynomial is of even degree, therefore the graph begins on the left above the x-axis. −2 is a root of even multiplicity, therefore at −2, the graph is tangent to the x-axis.
0 is a root of odd multiplicity, therefore 0 is a point of inflection.
3 is a root of odd multiplicity, therefore 3 is also a point of inflection.
Next Topic: Reflections and symmetry
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