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This 3 page Class Notes was uploaded by Edwina Flatley on Friday October 23, 2015. The Class Notes belongs to A at University of Kentucky taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/228263/a-university-of-kentucky in Studio Art at University of Kentucky.
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Date Created: 10/23/15
Seminar 6 Quadratics inequalities and absolute values longdivision remainders and factoring AS 101003 High school mathematics from a more advanced point of View 1 Welll continue With quadratic equations today the connection between nding roots7 graphs7 and factoring 2 We may see up to two presentations today 3 As time permits Why does factoring a function help nd roots of 4 Division stuff numbers and polynomials Quick review Problem 1 Let 41 12 How can you nd the minimum value of Can you do it Without graphing7 or even visualizationThat is7 argue you have the correct min using number facts Problem 2 Let z 7 22 5 How can you nd the minimum value of Can you do it Without graphing7 or even visualization Problem 3 Let hz l1 7 4 5 How can you nd the minimum value of Can you do it Without graphing7 or even visualization Problem 4 Make up realworld maximization or minimization problem modeled by a quadratic function Discussion problem 5 What s a root of a quadratic function What connection does a root a quadratic have With its graph Can you explain that connection Problem 6 Find the roots of 1 127312 2 91 712 4zl Problem 7 What connection does the factorization of a quadratic have to with its roots as a function For example let 41 12 7 312i Factor Determine its roots Problem 8 Suppose we have factored a quadratic 41 1 7 c1 7 d What are the roots of 41 Explain How can you be sure that we haven7t missed any roots Before you explain work on the next problem Problem 9 Let7s work in Z12 Let 1 7 2 1 7 4 Are the roots of just 2 and 4 Problem 10 What property of real numbers not possessed by Z12 guarantees that factoring completely determines the root set of a quadratic Absolute values and inequalities too because they are so important in high school math Let7s recall the 50 cent de nition of absolute value 1 if 1 2 0 otherwise 1 71 when 1 lt 0 Problem 11 Explain how to nd solutions to 2 Tol17ll2 Tol217ll720 Problem 12 Now you want to nd solutions to gt 1 How many solutions are there Can you provide a graphical description of the solution set Problem 13 Solve 21 7 1 2 3 Problem 14 Connect absolute values and the idea of distance on the number line Problem 15 How would you solve 12 7 4 2 0 Loose ends should we go here Please look this over this week I think some discussion of longdivision divisibility in numbers and polynomials might be interesting and useful ultimately to you The famous Division Algorithm so obvious it s hard to see77ubut having so many uses Let ab be integers with I f 0 Then there exist unique 4 the quotient and T the remainder with T satisfying 0 S T lt I such that a 124 T Let a 1001 and b 21 Find 4 and T guaranteed by the Division Algorithm Unless you re good at mental arithmetic you might do a longdivision One consequence of the uniqueness aspect of the Division Algorithm an integer is either even or odd and it can t be both Of course everyone knows that anywayuso of course its a consequence of a detestable theorem For that matter why does the longdivision algorithm on integers work anyway Simple explanation Ever try to nd one I didnltubut now l7m sorry I didn7t7 because I have to teach the validity of that algorithm to preservice elementary teachers which makes me feel sort of phony but relieved to nd out it s explainable in simple terms TheoTem Let 101 be a polynomial of degree n Then T is a root of 101 if and only if 101 I 7 Tqz where qz is a polynomial of degree n 7 1 What does this theorem say about the number of roots of a polynomial of degree n
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