# 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
• 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
• Slide 22
• Slide 23
• Given: What can you identify about this equation? 2 nd : Sum of the roots: Which means:
• Slide 24