Download PreCalculus Section 2.7 Apply general theorems about polynomials

PreCalculus Section 2.7
Apply general theorems about polynomials
• Fundamental Theorem of Algebra
In the complex number system, a polynomial of
degree n has exactly n roots.(cubic has three
roots, quartic has four roots, etc..)
How many roots does P(x) = 3x5 – 6x2 + 9 have?
Answer: five
Complex Conjugates Theorem
If a polynomial with real coefficients has the complex
root a + bi, then it also has a – bi as a root.
If 7 – 6i is a root of a polynomial with real
coefficients, what is another of its roots?
Answer: 7 + 6i
Irrational Conjugates Theorem
If a polynomial with rational coefficients has the irrational
root a + √b then it also has the root a - √b
If 3 + √2 is a root of a polynomial with rational
coefficients, then 3 - √2 is also a root.
Odd degree Theorem
If a polynomial has real coefficients and an odd degree,
then it has at least one real root.
For the equation
axn + bxn-1 + cxn-2 + … dx + e
The sum of the roots is
given by: -b/a
The product of the roots
is given by:
e/a if n is even
-e/a if n is odd
• Find the sum of the roots • Find the product of the roots
of 3x4 + 2x3 - 6x2 + 7x + 9
of 3x4 + 2x3 - 6x2 + 7x + 9
• Answer: -2/3
• Answer: 9/3 = 3
Quadratic Equation Theorem
• A quadratic equation can be written as:
• x2 – (sum of the roots)x +(product of the roots) = 0
• Find the equation of the quadratic equation with
roots of 3 + 2i and 3 – 2i.
• Find the equation of the cubic equation with roots
of 5 - √3, 5 + √3, and 6.
assignment
• Page 89
• Problems 1-10,13,14,18,19,20,24