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2.57
PART E: THE FUNDAMENTAL THEOREM OF ALGEBRA (FTA)
The Fundamental Theorem of Algebra (FTA)
If
f x
( )
is a nonconstant
n
th
-degree polynomial in standard form with real
coefficients, then it must have at least one complex (possibly real) zero.
Put Another Way: It must have exactly
n
complex zeros, where the zeros may be
repeated based on their multiplicities.
Technical Note: The Fundamental Theorem of Arithmetic
states that any integer greater
than or equal to 2 is either prime or can be decomposed uniquely as a product of
(possibly repeated) primes (or “prime powers”), up to a reordering of the factors.
For example, 6 can only be decomposed in one way:
6
=
2
⋅
3
. The decomposition
6
=
3
⋅
2
does not count as a different one.
Technical Note: The Fundamental Theorem of Calculus
will allow you to evaluate
definite integrals, which are used in finding areas, volumes, arc lengths, surface areas,
and much more.
Historical Note: The FTA was first proven by the great Gauss. For more history, see
p.193 of the textbook
.

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2.58
PART F: THE LINEAR FACTORIZATION THEOREM (LFT)
The Linear Factorization Theorem (LFT)
If
f x
( )
is a nonconstant polynomial in standard form with real coefficients,
then it must have a factorization into linear factors of the form:
f x
( )
=
a
n
x
−
c
1
( )
x
−
c
2
( )
x
−
c
n
( )
a
n
∈
R
;
a
n
≠
0; each
c
i
∈
C
( )
Note: The zeros of
f x
( )
are then
c
1
,
c
2
,
…
,
c
n
.
Note: There may be repetitions of a zero
c
i
, based on the multiplicity of
c
i
.
Note:
a
n
is the leading coefficient of
f x
( )
.
Technical Note: The LFT is proven using the FTA and the Factor Theorem.
See
p.193 of the textbook
.
Technical Note: This helps explain the Complex Conjugate Pairs Theorem in
Notes 2.56
.
Example
Let
f x
( )
=
x
5
−
8
x
4
+
16
x
3
.
x

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