5.7 Apply the Fundamental Theorem of Algebra

1
5.7 Apply the Fundamental Theorem of Algebra
2
Here is the graph of f(x). How many real zeros
do you see? What are their values?
This third degree equation has 3 zeros x 3,
x -2, and x -3
3
Can be factored as
Here is the graph of f(x).
How many times is f(x)0?
Since (x-5) is a factor twice (notice the
exponent), we say x 5 is a repeated solution.
We say this third degree equation also has 3
zeros x -4, and x 5 and 5.
4
The equation
Has three solutions x -3 , x 4i, and x 4i.
And thus 3 factors (x3)(x-4i)(x4i)
Has four zeros -2, -2, -2 and 0
And thus four factors (x2)(x2)(x2)(x) or
5
The following important theorem, called the
fundamental theorem of algebra was first proved
by the famous German mathematician Carl Friedrich
Gauss (1777-1855).
Fundamental Theorem of Algebra If f(x) is a
polynomial of degree n where n gt 0, then the
equation f(x) 0 has at least one root in the
set of complex numbers.
The important consequences of this theorem In
general, when all real and imaginary solutions
are counted (with all repeated solutions counted
individually), any n th degree polynomial
equation has exactly n solutions. Similarly, any
n th-degree polynomial has exactly n zeros.
6
Find all the zeros of
Solutions
The possible rational zeros are
Synthetic division or the graph can help
Notice the real zeros appear as x-intercepts. x
1 is repeated zero since it only touches the
x-axis, but crosses at the zero x -2.
Thus 1, 1, and 2 are real zeros. Find the
remaining 2 complex zeros.
7
Find all the zeros of
Solutions
The zeros are 1, 1, -2, and
Thus
Notice the complex zeros occurred in a conjugate
pair.
The complex zeros of a polynomial functions with
real coefficients always occur in complex
conjugate pairs. That is, if a bi is a zero,
then a bi must also be a zero.
8
Write a polynomial function f of least degree
that has real coefficients, a leading coefficient
1, and 2 and 1 i as zeros.
Solution
f(x) (x 2)x (1 i)x (1 i)
9
The rational zero theorem gives you a way to find
the rational zeros of a polynomial function with
integer coefficients. To find the real zeros of
any polynomial functions, you may need to use a
calculator.
For example, approximate the zeros of
From the screens you can see the real zeros are
about 0.73 and 2.73