Identifying Points of Discontinuity in a Polynomial Function
Explore how to determine points of discontinuity in a polynomial function on a given interval. Discover roots of equations and understand the concept of polynomial functions. Practice finding zeros by completing the square and expanding perfect squares. Evaluate functions and discover patterns in perfect squares expansions.
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Presentation Transcript
Bell work: Turn in when completed Determine any points where the function is not continuous on the interval [-5, 5] 7 ? ?(?) = 5(?+5)(? 2)
4.1: Polynomial Functions
A polynomial function is any function of a single variable involving only constant coefficients and powers of that variable; i.e.:
Degree: Leading Coefficient: Zeros: Roots:
Find the roots of the following equation ? ? = ?2+ 5? + 6
Find the roots of the following equation
Find the roots of the following equation ? ? = ?2+ 4? + 5
Find the roots of the following equation by completing the square ? ? = ?2+ 5? + 6
Find the zeros for the function below by completing the square ? ? = ?2+ 4? 11
Find the zeros for the function below by completing the square ? ? = 2?2+ 6? 8
Find the zeros for the function below by completing the square ? ? = 3?2+ 7? + 7
Group Bell Work: Expand the following perfect squares. Look for any patterns you can find. (? + 3)2 (? + 5)2 (? + 10)2
Bell work: Evaluate the function ? ? =2+3? 5+5? Find ?(?3+ 1)
Write a polynomial with roots 15 and -277 How many times does this polynomial cross the x-axis?
Fundamental Theorem of Algebra Every polynomial equation with degree greater than 0 has at least one root in the set of complex numbers Therefore, every polynomial of degree n can be written as a product of n factors
Properties of Imaginary Numbers Powers of i Complex Numbers Conjugates
Write a polynomial with roots 5, 15, 22, 15 How many times does this polynomial cross the x-axis?
Write a polynomial with roots 3, i, and -i How many times does this polynomial cross the x-axis?
How many complex roots does the following equation have?
Group work: Turn in when complete Find the roots of the equations below by completing the square 1. 0 = 12?2+ 6? + 2 2. 0 = ??2+ 6? + 2 3. 0 = ??2+ ?? + 2 4. 0 = ??2+ ?? + ?
Determining location of zeros using the discriminant If ?2 4?? < 0, then two imaginary roots If ?2 4?? = 0, then one real root If ?2 4?? > 0, then two real roots
Complex Conjugates Theorem If a polynomial has a root of the form ? + ??, then it must also have a root of the form ? ??.
Homework: Completing the Square Worksheet (Done) Chapter 4.2, pages 219-220: 20- 26, 28, 31, 35
