Download Important Theorems for Algebra II and/or Pre

Important Theorems for Algebra II and/or Pre-Calculus
State the following theorems and provide an illustration of each theorem. Do each theorem and an
illustration/example on exactly one side of a sheet of notebook paper. You may use both the front side and
backside of the notebook paper, but only one theorem may be completed per side. Your
illustration/example should be brief and to the point, not just one part in the solution of a long problem.
1.
2.
3.
4.
5.
6.
7.
8.
Fundamental Theorem of Algebra (FTA)
Rational Root Theorem (RRT)
Complex Conjugate Root Theorem (CCRT)
Factor Theorem
Remainder Theorem
Discriminant Theorem
Descartes' Rule of Signs
Sum and Product of Roots Theorem
← called Nature of Solutions
← Theorem 5 in our text
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Since these theorems are worded differently, and often poorly, in different texts, I am providing you with
my preferred version of each theorem. You may cite a different version if you prefer.
1.
Fundamental Theorem of Algebra (FTA)
Part (a) : Statement of the Theorem
Fundamental Theorem of Algebra (FTA)
Every polynomial equation with positive degree n has exactly n complex roots.
Note that for this theorem, a double root counts as 2 roots, a triple root counts as
three roots, and so on.
Part (b) : Illustration/Example
By the FTA, the equation x
3
= 8 has 3 solutions (over the complex numbers of course).
2.
Rational Root Theorem (RRT)
Part (a) : Statement of the Theorem
Rational Root Theorem (RRT)
bg
Let P x be a polynomial with integral coefficients and a nonzero constant term.
b g = 0 has a rational root
p
that is in lowest terms,
q
then p must be an integral factor of the constant term, and q must be an integral
If the polynomial equation P x
factor of the leading coefficient.
Part (b) : Illustration/Example
(for you to do)
3.
Complex Conjugate Root Theorem (CCRT)
Part (a) : Statement of the Theorem
Complex Conjugate Root Theorem (CRRT)
If a polynomial equation with real coefficients has a + bi as an imaginary root,
then a − bi is also a root.
Part (b) : Illustration/Example
(for you to do)
4.
Factor Theorem
Part (a) : Statement of the Theorem
Factor Theorem
bg
b g = 0,
For the polynomial P x and the polynomial equation P x
x − r is a factor if and only if
r is a root
Part (b) : Illustration/Example
(for you to do)
5.
Remainder Theorem
Part (a) : Statement of the Theorem
Remainder Theorem
bg
bg
When a polynomial P x is divided by x − a , the remainder is P a .
Part (b) : Illustration/Example
(for you to do)
6.
Discriminant Theorem
Part (a) : Statement of the Theorem
Discriminant Theorem
Given the quadratic equation ax + bx + c = 0 , where a , b , and c are real numbers:
2
If b − 4ac > 0 , there are two unequal real roots.
2
If b − 4ac < 0 , there are two unequal imaginary roots (called complex conjugates).
2
If b − 4ac = 0 , there is one real root (called a double root).
2
Part (b) : Illustration/Example
(for you to do)
7.
Descartes' Rule of Signs
Part (a) : Statement of the Theorem
Descartes' Rule of Signs
bg
If P x is a polynomial with real coefficients, then
1.
bg=0
the number of positive roots of P x
bg
is either equal to the number of
variations of sign of P x or is less than this number by a positive even integer ;
2.
bg=0
the number of negative roots of P x
b g
is either equal to the number of
variations of sign of P − x or is less than this number by a positive even integer .
Part (b) : Illustration/Example
Here's a good equation to say something immediate about:
8.
x 4 + 2 x 3 − 2 x 2 − 6x − 3 = 0
Sum and Product of Roots Theorem
Part (a) : Statement of the Theorem
Sum and Product of Roots Theorem
For the equation an x + an −1 x
n
n −1
+ an − 2 x n − 2 + K + a1 x + a0 = 0 , with an ≠ 0 :
an − 1
the sum of the roots is
−
the product of the roots is
R| a
|S a
|| − a
T a
an
0
n
0
n
Part (b) : Illustration/Example
(for you to do)
if n is even
if n is odd