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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