Alg - Loudoun County Public Schools
Name: Date: Block: The Fundamental Theorem of AlgebraFundamental Theorem of AlgebraYou have solved quadratic equations and found that their roots are included in the set of complex numbers. In other words, the roots have been integers, rational numbers, irrational numbers, and imaginary numbers. But, the question remains, can all polynomial equations be solved using complex numbers?1579547014400Carl Friedrich Gauss (1777-1855) proved that the answer to this question is yes! The roots of every polynomial equation, even those with imaginary coefficients, are complex numbers. The answer to this question was so important that it is now known as the Fundamental Theorem of Algebra. Definition of the Fundamental Theorem of AlgebraIf P(x) is a polynomial of degree n ??1 with complex coefficients, then P(x) ??0 has at least one complex root.The Fundamental Theorem of Algebra concludes that including imaginary roots and multiple roots, an n-th degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros.Exploration: A polynomial can have, at most, the same number of real roots as its highest degree. Last unit we have worked quadratics that had either , , or real roots. If a quadratic function had no real roots we said that this function had roots.NOTE: Imaginary roots of any polynomial must be a complex set…in other words imaginary roots must be in pairs. For example, if a polynomial has a root of , then the second root that is paired with this is .Example #1 – Let’s look at three different functions & determine the combinations of real and non-real roots. Quadratic Cubic Quartic47169241093638Real RootsNon-Real, Complex Roots00Real RootsNon-Real, Complex Roots24826821102264Real RootsNon-Real, Complex Roots00Real RootsNon-Real, Complex Roots3605841128143Real RootsNon-Real, Complex Roots00Real RootsNon-Real, Complex Roots Application of the FTAExample #2 – Given the function . a) Determine the degree and leading coefficient. b) Determine the end behavior of c) What are the zeros of Example #3 – Given the function .a) Determine the degree and leading coefficient.b) What are the zeros of c) At which zero(s) will the graph of cross through the x-axis?d) At which zero(s) will the graph of hit the x-axis but not cross through?Definition of MultiplicityThe multiplicity of a graph refers to the that its associated factor appears in the polynomial function.Example #4 – Given , determine the zeroes and the multiplicity of each zero.NotesThe multiplicity of a zero determines whether the graph crosses the x-axis at that zero or if it instead turns back the way it came (bounced off the x-axis).If a zero has an odd multiplicity then the graph will cross through the x-axis at that zero. Additionally, if a zero has an even multiplicity then the graph will not cross through (bounced off) the x-axis at that zero.Example #5 – Each of the following graphs have zeros at and Write the equation for each by determining the multiplicity of each root.96499211303000410400514272000 GRAPH A GRAPH B GRAPH C GRAPH D10540626985004071705665900Example #6 – Without using a calculator, write and then sketch the graph of a least degree polynomial using the information provided. a) has a negative leading coefficient and b) has a positive leading coefficient zeros of with multiplicity 2 and and zeros of with multiplicity 3, 8004518572500 and with multiplicity 2.42763652349500Rational Root Theorem82711182425p is a factor of and q is a factor of .020000p is a factor of and q is a factor of .Given a polynomial f(x) the only possible rational solutions of the equation f(x) = 0 are . WhereExample #7 – List all possible rational zeros of the given functions using the rational root theorem.a) b) NOTE: These are only possibilities – we will investigate in the future to see which ones (if any) ARE zeros for the function! ................
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