General Results for Polynomial Equations

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Presentation transcript:

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

Objective:

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

The Fundamental Theorem of Algebra:

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).

The Complex Conjugates Theorem:

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.

Irrational Roots Theorem:

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.

Odd Degree Polynomial Theorem:

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!

Theorem 5:

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

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

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

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

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

Given:

Given: What can you identify about this equation?

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

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

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

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

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

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

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

Assignment: Pgs. 89 - 90 1 – 27 odd