Lesson 2-7 General Results for Polynomial Equations.

Lesson Lesson 2-72-7

General Results General Results for Polynomial for Polynomial

EquationsEquations

Objective:Objective:

Objective:Objective:

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

GivenGiven::

GivenGiven::

What can you identify about this equation?

GivenGiven::

What can you identify about this equation?

1st: Because this is an odd polynomialit has at least one real root.

GivenGiven::

What can you identify about this equation?

2nd: Sum of the roots:

GivenGiven::

What can you identify about this equation?

2nd: Sum of the roots:

GivenGiven::

What can you identify about this equation?

2nd: Sum of the roots:

Which means:

GivenGiven::

What can you identify about this equation?

3rd: Product of the roots:

GivenGiven::

What can you identify about this equation?

3rd: Product of the roots:

GivenGiven::

What can you identify about this equation?

3rd: Product of the roots:

Which means:

Assignment:Assignment:

Pgs. 89 - 90 Pgs. 89 - 90 1 – 27 odd1 – 27 odd