WARM UP- FOR THE POLYNOMIAL LISTED BELOW:. FUNDAMENTAL THEOREM OF ALGEBRA.

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Presentation transcript:

WARM UP- FOR THE POLYNOMIAL LISTED BELOW:

FUNDAMENTAL THEOREM OF ALGEBRA

LEARNING TARGETS  Introduce the FTA  Use FTA to determine how many solutions a polynomial will have  Look at imaginary vs. real solutions  Determine the minimum number of real solutions polynomials will have

DETERMINE A FACTORED FORM FOR THE FOLLOWING POLYNOMIAL ***The highest degree a term can be is cubed.

CONT.

DETERMINE A FACTORED FORM FOR THE FOLLOWING POLYNOMIAL ***The highest degree a term can be is cubed. What is the leading term? How many roots does this polynomial have?

SOLUTION

ANSWER THE FOLLOWING

IS THIS ALWAYS TRUE? Leading Term Power:Roots: n

FUNDAMENTAL THEOREM OF ALGEBRA  The fundamental theorem of algebra states that for each polynomial with complex coefficients there are as many roots as the degree of the polynomial.  If a polynomial’s leading term has degree n then it will have n roots.

FTA  Based on the FTA we must always have the same number of roots as our leading terms degree.  However, this does not mean all of our roots will be real numbers (where the graph crosses the x-axis).  Can you think of any potential functions that do not cross the x-axis the same number of times as their leading terms degree?  When this happens we have what are called imaginary roots (more on these next week)

FTA  The combination of real roots and imaginary roots must always equal the same degree of the leading term  Next we will view some graphs and look at the number of real roots and imaginary roots  While we look at these graphs pause and ponder and come up with some skeptical questions or I notice statements…

2 ND DEGREE POLYNOMIAL

3 RD DEGREE POLYNOMIAL

4 TH DEGREE POLYNOMIAL

5 TH DEGREE POLYNOMIAL

N TH DEGREE POLYNOMIAL

FTA RULES WITH POLYNOMIALS  Can an even degree polynomial have less than one real root? Why?  Can an even degree polynomial ever have an odd number of real roots? Why?  Can an odd degree polynomial have less than one real root? Why?  Can an odd degree polynomial ever have an even number of real roots? Why?

RECAP  Fundamental Theorem of Algebra:  Nth degree polynomials have N number of complex roots  A complex root can be real or imaginary  Real root occurs when the graph crosses the x-axis  The degree of a polynomial will determine the lowest amount of real roots that polynomial can have  This depends on odd-even and its end behavior

FOR TONIGHT:  Worksheet  I will be collecting these on Thursday and giving feedback

EXIT TICKET  Sketch the following graphs:  5 th Degree polynomial with a negative leading coefficient:  5 real roots  4 real roots  3 real roots  2 real roots  1 real root  0 roots  If the graph cannot be sketched put down N/A and state why