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understand what is meant by the multiplicity of a root of a polynomial, sketch the graph of a polynomial, given its expression as a product of linear factors. Contents. Introduction. What is a polynomial? 2. 3. Graphs of polynomial functions. 3. 4. Turning points of polynomial functions 6. 5. Roots of polynomial functions. 7. Introduction.
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- OBJECTIVES
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- Shape of the Graph of a Polynomial Function Near a Zero of Multiplicity k
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- Analyze the graph to address the following about the polynomial function it represents
Understanding the Definition of a Polynomial Sketching the Graphs of Power Functions Determining the End Behavior of Polynomial Functions Determining the Intercepts of a Polynomial Function Determining the Real Zeros of Polynomial Functions and Their Multiplicities Sketching the Graph of a Polynomial Function Determining a Possible Equation of a Po...
Use the end behavior of each graph to determine whether the degree odd and whether the leading coefficient is positive or negative. Degree: odd Leading coefficient: negative Degree: even Leading coefficient: positive
Suppose c is a real zero of a polynomial function f of multiplicity k (where k is a positive integer); that is, (x – c)k is a factor of f. Then the shape of the graph of f near c is as follows: Functions and Their Multiplicities Shape of the Graph of a Polynomial Function Near a Zero of Multiplicity k Suppose c is a real zero of a polynomial funct...
Find all the real zeros, determine the multiplicities of each zero, and decide whether the graph touches or crosses at each zero.
Determine the end behavior. Odd, with leading -, so left hand up, right hand down.
Determine the end behavior. Even, with leading +, so both ends up
Completely factor f to find all real zeros and their multiplicities.
a. Is the degree of the polynomial function even or odd? Odd because the graph starts up and finishes down. b. Is the leading coefficient positive or negative? Negative because the right hand behavior finishes down
rategy for Graphing Polynomial Functions) Find x-inte. the left-most and the right-most x-axis) Determine the multiplicity: for each x-intercept, se TOUCH or CROSS to fill in the graph.) Fi. Ex. Use knowledge of end behavior and multiplicity of zero to sketch the function: x3 x 2 . 2.
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The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor, x = 2, has multiplicity 2 because the factor (x − 2) occurs twice. The x -intercept −1 is the repeated solution of factor (x + 1)3 = 0.
Four-Step Process for Sketching the Graph of a Polynomial Function . 1. Determine the end behavior. 2. Plot the y-intercept . fa(0) = 0. 3. Completely factor f to find all real zeros and their multiplicities*. 4. Choose a test value between each real zero and sketch the graph.
Aug 15, 2024 · Write a formula for the polynomial function shown in Figure 3.2.20. Figure 3.2.20. Solution. This graph has three x-intercepts: x = − 3, 2, and 5. The y-intercept is located at (0, 2).At x = − 3 and x = 5, the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear.
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Definition 4 If a is a root of f, its multiplicity is the number of times the factor (x − a) appears in the factored form of f. For example, in the polynomial (x − 1)2, the root 1 has multiplicity 2. So even though it has one root, it will be useful to count the root 1 twice.
