Section 2.7 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. What is the Fundamental Theorem of Algebra? Where do we use the Fundamental Theorem of Algebra?. German mathematician Carl Friedrich Gauss (1777-1855) first proved this theorem. The Fundamental
Theorem of Algebra
If f(x) is a polynomial of degree n
where n > 0, then
the equation f(x) = 0 has at least one root in the set of complex numbers.
If f(x)is a polynomial of degree nwhere
n >0, then the equation f(x) = 0has
exactly nsolutions provided that
each solution repeated twice
is counted as 2 solutions, etc…
solve each of the following polynomial equation, proved this theorem.
determine how many solutions each equation has,
and classify each solution as rational, irrational or imaginary.
2x− 1 = 0
Solution is: x = ½, one, rational
x2 − 2 = 0
- two, irrational
x3 − 1 = 0
Solution: (x−1)(x2 + x + 1), x = 1 – one rational, and use Quadratic formula for
So, how many solutions do following equations have? Find all zeros and classify them:
x3 + 3x2 + 16x + 48 = 0
(x + 3)(x2 + 16)= 0
x + 3 = 0, x2 + 16 = 0
x = −3, x2 = −16
x = − 3, x = ± 4i,
1 rational, 2 imaginary – 3 total
x4 + 5x2 - 36 = 0
(x2 - 4) (x2 + 9) = 0
x2 – 4 = 0, x2 + 9 = 0
x2 = 4, x2= − 9
x = ± 2, x = ± 3i,
2 rational, 2 imaginary – 4 total
Next- write the complete factorization
of this polynomial function
Real zeros → can see on graph as x-intercepts
Imaginary zeros → cannot see on graph
ALSO, the graph is tangent to the x-axis at the repeated zero x = − 1, but crosses the x-axis at the zero x = 2.
Are there any pattern??
When a factor x – k of a function f is raised to an odd power,
the graph of f crosses the x-axis at x = k.
When a factor x – kof a function f is raised to an even power,
the graph of f is tangent (bounces off) to the x-axis at x = k.
Complex zeros have? Find all zeros and classify them:always occur in conjugate pairs
“If a + bi is a zero, then a – bi is also”
This is The Complex Conjugates Theorem:
If f is a polynomial function with real coefficients,
and a + bi is an imaginary zero of f,
then a – bi is also a zero of f.
Irrational Conjugates Theorem:
If f is a polynomial function with rational coefficients,
and is a zero of f,
then is also a zero of f.
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros:
Are there any other ways to find the number
of positive and negative real zeros?
Let f(x) be a polynomial function with real coefficients.
This rule works even if some terms are missing!
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.
Possible rational zeros theorem (PRZ), the graph appearance, synthetic division, and quadratic formula.
If x = #, it becomes a factor (x ± #). Multiply factors together to find the equation.
Group work: have? Find all zeros and classify them:
Open your book on pg. 142
and work in pairs to complete
## 16, 18, 26, 28, 35 – 49 odd
Section 2.8 have? Find all zeros and classify them:
Analysis and Graphing of Polynomial Functions
HOW to do it WITHOUT
Open your book on pg. 147 and work with your partner on ## 1 – 4 from “Guided practice” at the bottom of the page
Maximizing and minimizing
in problem solving….
Read Ex. 3 on pg. 147 and complete # 39 on pg. 149