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:
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::
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