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RF3: Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).
Functions and Relations
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RF4: Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5).
Functions and Relations
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Note:
- n is a whole number- x is a variable- the coefficients an to a0 are real numbers
Each exponent of a polynomial must be a whole number defined as {0, 1, 2, 3, ...}.
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The Shape of a Polynomial
The graph of a polynomial function is continuous. This means that you can draw the complete graph without lifting your pencil from the paper.
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A polynomial has no breaks or sharp corners.
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End Behavior of Polynomials
The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction. To describe end behavior, we can use the following notation:
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End Behavior of Polynomials
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Peaks and Valleys(Turning Points/BUMPS/Extreme Points)
For degree n, a polynomial graph will have, at most, n-1 bumps.
Peak/Turning point/bump
Valley/Turning point/bump
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Peaks and Valleys(Turning Points/BUMPS/Extreme Points)
For degree n, a polynomial graph will have, at most, n-1 peaks and valleys.
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Sample ProblemWhat is the minimum possible degree of the polynomial graphed below?
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Since there are four bumps on the graph, and since the end-behavior says that this is an odd-degree polynomial, then the degree of the polynomial is 5, 7, or 9...
Therefore, the minimum possible degree is 5.
Answer Key
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Zeros, Roots and X-Intercepts
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a) Determine if each given value is a zero of P(x) = x2 4x 5.
i) x = 1 ii) x = 3
Sample Problem
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b) Find the zeros of P(x) = x2 4x 5 by solving for the roots of the related equation, P(x) = 0.
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c) Use a graphing calculator to graph P(x) = x2 4x 5.
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Answer Key
P(x) = x2 4x 5
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Multiplicity of a Zero
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We will be working with polynomial functions that have a multiplicity of 1, 2 or 3.
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For the following graphs determine the zeros and state each zero's multiplicity.
Check Your Understanding
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Zeros Multiplicity
a)
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Zeros Multiplicity
b)
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Zeros Multiplicity
c)
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Zeros Multiplicity
d)
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Sample Problems
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Answer Key
a)
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Answer Key
b)
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Check Your Understanding
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Answer Key
a)
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b)
Answer Key
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Writing an Equation from a Graph
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Sample Problem
Determine the polynomial function corresponding to the graph. You may leave your answer in factored form.
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Answer Key
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Sample Problem
Determine the polynomial function corresponding to the graph. You may leave your answer in factored form.
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Answer Key
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Sample Problem
Given the characteristics of a polynomial,
draw the graph and derive the actual function.
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Answer Key
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Sample Problem
Given the characteristics of a polynomial,
draw the graph and derive the actual function.
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Answer Key