Fundamental Theorem of Algebra

Warm Up- For the polynomial listed below:

Fundamental Theorem of Algebra

Learning TargetsIntroduce the FTAUse FTA to determine how many solutions a polynomial will haveLook at imaginary vs. real solutionsDetermine 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 beis cubed.What is the leading term?

How many roots does this polynomial have?

SolutionAnswer the FollowingIs this always true?Roots:1234nFundamental Theorem of AlgebraThe fundamental theorem of algebra states that for each polynomial with complexcoefficientsthere are as manyrootsas thedegree of the polynomial.

If a polynomials leading term has degree n then it will have n roots. FTABased 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)FTAThe 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 statements2nd Degree Polynomial

3rd Degree Polynomial

3rd Degree Polynomial

4th Degree Polynomial

4th Degree Polynomial

5th Degree Polynomial

nth Degree Polynomial

Read List of Examples for Students to Solve

19FTA Rules with PolynomialsCan 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?RecapFundamental Theorem of Algebra:Nth degree polynomials have N number of complex roots

A complex root can be real or imaginaryReal 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 haveThis depends on odd-even and its end behavior

For tonight:WorksheetI will be collecting these on Thursday and giving feedbackExit TicketSketch the following graphs:5th Degree polynomial with a negative leading coefficient:5 real roots4 real roots3 real roots2 real roots1 real root0 roots

If the graph cannot be sketched put down N/A and state why