1. General Results for Polynomial Equations
Lesson 2-7
General Results for Polynomial Equations

2. Objective:

3. To apply general theorems about polynomial equations.
Objective:
To apply general theorems about polynomial equations.

4. The Fundamental Theorem of Algebra:

5. The Fundamental Theorem of Algebra:
In the complex number system consisting of all real and imaginary numbers, if P(x) is a polynomial of degree n (n>0) with complex coefficients, then the equation P(x) = 0 has exactly n roots (providing a double root is counted as 2 roots, a triple root as 3 roots, etc).

6. The Complex Conjugates Theorem:

7. The Complex Conjugates Theorem:
If P(x) is a polynomial with real coefficients, and a+bi is an imaginary root, then automatically a-bi must also be a root.

8. Irrational Roots Theorem:

9. Irrational Roots Theorem:
Suppose P(x) is a polynomial with rational coefficients and a and b are rational numbers, such that √b is irrational. If a + √b is a root of the equation P(x) = 0 then a - √b is also a root.

10. Odd Degree Polynomial Theorem:

11. Odd Degree Polynomial Theorem:
If P(x) is a polynomial of odd degree (1,3,5,7,…) with real coefficients, then the equation P(x) = 0 has at least one real root!

12. Theorem 5:

13. Theorem 5: For the equation axn + bxn-1 + … + k = 0,
with k ≠ 0 the sum of roots is:

14. Theorem 5: For the equation axn + bxn-1 + … + k = 0,
with k ≠ 0 the sum of roots is:

15. Theorem 5: For the equation axn + bxn-1 + … + k = 0,
with k ≠ 0 the product of roots is:

16. Theorem 5: For the equation axn + bxn-1 + … + k = 0,
with k ≠ 0 the product of roots is:

17. Theorem 5: For the equation axn + bxn-1 + … + k = 0,
with k ≠ 0 the product of roots is:

18. Given:

19. Given:
What can you identify about this equation?

20. Given: What can you identify about this equation?
1st: Because this is an odd polynomial
it has at least one real root.

21. Given: What can you identify about this equation?
2nd: Sum of the roots:

22. Given: What can you identify about this equation?
2nd: Sum of the roots:

23. Given: What can you identify about this equation?
2nd: Sum of the roots:
Which means:

24. Given: What can you identify about this equation?
3rd: Product of the roots:

25. Given: What can you identify about this equation?
3rd: Product of the roots:

26. Given: What can you identify about this equation?
3rd: Product of the roots:
Which means:

31. Assignment: Pgs – 27 odd