# lesson 2-7 general results for polynomial equations

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- Slide 1
- Lesson 2-7 General Results for Polynomial Equations
- Slide 2
- Objective:
- Slide 3
- Objective: To apply general theorems about polynomial equations.
- Slide 4
- The Fundamental Theorem of Algebra:
- Slide 5
- 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).
- Slide 6
- The Complex Conjugates Theorem:
- Slide 7
- 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.
- Slide 8
- Irrational Roots Theorem:
- Slide 9
- 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.
- Slide 10
- Odd Degree Polynomial Theorem:
- Slide 11
- 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!
- Slide 12
- Theorem 5:
- Slide 13
- For the equation ax n + bx n-1 + + k = 0, with k 0 the sum of roots is:
- Slide 14
- Theorem 5: For the equation ax n + bx n-1 + + k = 0, with k 0 the sum of roots is:
- Slide 15
- Theorem 5: For the equation ax n + bx n-1 + + k = 0, with k 0 the product of roots is:
- Slide 16
- Theorem 5: For the equation ax n + bx n-1 + + k = 0, with k 0 the product of roots is:
- Slide 17
- Theorem 5: For the equation ax n + bx n-1 + + k = 0, with k 0 the product of roots is:
- Slide 18
- Given:
- Slide 19
- What can you identify about this equation?
- Slide 20
- Given: What can you identify about this equation? 1 st : Because this is an odd polynomial it has at least one real root.
- Slide 21
- Given: What can you identify about this equation? 2 nd : Sum of the roots:
- Slide 22
- Given: What can you identify about this equation? 2 nd : Sum of the roots:
- Slide 23
- Given: What can you identify about this equation? 2 nd : Sum of the roots: Which means:
- Slide 24
- Given: What can you identify about this equation? 3 rd : Product of the roots:
- Slide 25
- Given: What can you identify about this equation? 3 rd : Product of the roots:
- Slide 26
- Given: What can you identify about this equation? 3 rd : Product of the roots: Which means:
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Assignment: Pgs. 89 - 90 1 27 odd

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