2. The Fundamental Theorem of Algebra 1. What is the Fundamental Theorem of Algebra? 2. Where do we use the Fundamental Theorem of Algebra?

3. German mathematician Carl Friedrich Gauss (1777-1855) first proved this theorem. If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers. The Fundamental Theorem of Algebra

4. Other wording…… If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided that each solution repeated twice is counted as 2 solutions, etc…

5. LETS 1)solve each of the following polynomial equation, 2)determine how many solutions each equation has, 3)and classify each solution as rational, irrational or imaginary. Ex. 1 2x − 1 = 0Solution is: x = ½, one, rational Ex. 2x 2 − 2 = 0Solution: - two, irrational Ex. 3 x 3 − 1 = 0Solution: (x −1)(x 2 + x + 1), x = 1 – one rational, and use Quadratic formula for - two, imaginary

6. So, how many solutions do following equations have? Find all zeros and classify them: (x + 3)(x 2 + 16)= 0 x + 3 = 0, x 2 + 16 = 0 x = −3, x 2 = −16 x = − 3, x = ± 4i, 1 rational, 2 imaginary – 3 total 1.x 3 + 3x 2 + 16x + 48 = 0 2.x 4 + 5x 2 - 36 = 0 (x 2 - 4) (x 2 + 9) = 0 x 2 – 4 = 0, x 2 + 9 = 0 x 2 = 4, x 2 = − 9 x = ± 2, x = ± 3i, 2 rational, 2 imaginary – 4 total

7. Find all the zeros of the function by applying PRZs, Factoring, and Quadratic formula: Zeros are: Next - write the complete factorization of this polynomial function

8. NOW, lets graph this function using your calculator!!! What do you observe??? ALSO, the graph is tangent to the x-axis at the repeated zero x = − 1, but crosses the x-axis at the zero x = 2. Are there any pattern?? Real zeros → can see on graph as x-intercepts Imaginary zeros → cannot see on graph YES! When a factor x – k of a function f is raised to an odd power, the graph of f crosses the x-axis at x = k. When a factor x – k of a function f is raised to an even power, the graph of f is tangent (bounces off) to the x-axis at x = k.

9. Complex zeros always occur in conjugate pairs “If a + bi is a zero, then a – bi is also” This is The Complex Conjugates Theorem: If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of f, then a – bi is also a zero of f.

10. How about irrational zeros? Irrational Conjugates Theorem: If f is a polynomial function with rational coefficients, and is a zero of f, then is also a zero of f.

11. How can we use the FTA to create the polynomial? Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: 1. 2.

12. Descartes' Rule of Signs Let f(x) be a polynomial function with real coefficients. The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number. The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number. Are there any other ways to find the number of positive and negative real zeros? This rule works even if some terms are missing!

13. Determine the number of possible positive real zeros, negative real zeros, and imaginary zeros for f(x) PRACTICE!

14. Lets summarize what we have learned: What is the Fundamental Theorem of Algebra? If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers. What methods do you use to find the zeros of a polynomial function? Possible rational zeros theorem (PRZ), the graph appearance, synthetic division, and quadratic formula. How do you use zeros to write a polynomial function? If x = #, it becomes a factor (x ± #). Multiply factors together to find the equation.

15. Group work: Open your book on pg. 142 and work in pairs to complete ## 16, 18, 26, 28, 35 – 49 odd Group work: Open your book on pg. 142 and work in pairs to complete ## 16, 18, 26, 28, 35 – 49 odd

16. Section 2.8 Analysis and Graphing of Polynomial Functions HOW to do it WITHOUT a calculator???

17. Steps for Graphing of Polynomial Functions 1. Find x-intercepts (zeros of the function) 2. Find y-intercept. 3. Determine end behavior (leading coefficient test) 4. Plot points between and beyond the x-intercepts and y-intercept 5. Draw the graph trough the plotted points so it has the appropriate end behavior.

18. Special Points on the graph – turning points They are a local maximum or a local minimum A polynomial of degree n has at most n – 1 turning points If a polynomial of degree n has n distinct real zeros, then the graph has exactly n – 1 turning points To find the turning points with the graphing calculator use maximum & minimum in CALC menu

19. Practice time! Open your book on pg. 147 and work with your partner on ## 1 – 4 from “Guided practice” at the bottom of the page

20. When do we use turning points of a polynomial in real life?? Maximizing and minimizing in problem solving…. Read Ex. 3 on pg. 147 and complete # 39 on pg. 149