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College Algebra: Lesson 5.4 The Fundamental Theorem of Algebra
Recall: An n th degree polynomial can have at MOST n real zeros.
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In the complex number system, this can be changed to an n th degree polynomial function has exactly n zeros.
The Fundamental Theorem of Algebra: If f(x) is a polynomial of degree n, where n > 0, then f has at least one
zero in the complex number system.
(Remember: Real numbers are a subset of the Complex Numbers!)
The Linear Factorization Theorem: If f(x) is a polynomial of degree n, where n > 0, then f has precisely n factors
f ( x)  an ( x  c1 )( x  c2 )    ( x  cn ) , where c1 , c2 , … , cn are complex numbers.
These theorems are called existence theorems, because they don’t tell you HOW to find the zeros!
If a factor is repeated, it is said to have a multiplicity of the value of the exponent on the factor. If there is an
exponent, the graph will "flatten out" at that zero. Also, if the multiplicity is odd, the graph crosses the x-axis at
that zero. If the multiplicity is even, the graph has a turning point at that zero.
Conjugate Pairs: Complex zeros occur in conjugate pairs. Let f (x ) be a polynomial function that has real
coefficients. If a + bi, where b  0 , is a zero of the function, the conjugate a – bi is also a zero of the function.
Examples:
Step 1 of 3: Determine the degree and enter the y-intercept as an ordered pair. Step 2 of 3: Determine the xintercept(s) at which f crosses the axis. (First select the number of x-intercept(s) at which f crosses the axis, then enter
the ordered pair(s).) Step 3 of 3: Determine the zero(s) of f at which it "flattens out". (First select the number of
zero(s), then enter the ordered pairs.
1. f ( x)  ( x  1)( x  2) 5
2. f ( x)  ( x  1) 2 ( x  4) 3
Use all available methods (e.g. the Rational Zero Theorem, Descarte's Rule of Signs, polynomial division, etc.) to find the
zeros of the following polynomial function. Use the linear factors to make sure you find the appropriate number of
solutions, counting multiplicity. Note: Zeros of multiplicity more than one only need to be entered once.
1. f ( x)  x 3  19 x 2  121x  259
2. f ( x)  x 5  4 x 4  30 x 3  102 x 2  125 x  50
3. Construct a polynomial with the following properties: fifth degree, 2 is a zero of multiplicity 2, -2 is the only other
zero, leading coefficient is 3.
4. Use all available methods (in particular, the Conjugate Roots Theorem, if applicable) to factor the polynomial function
completely, making use of the given zero.
a. f ( x)  x 4  12 x 3  81x 2  588 x  1568 ; -7i is a zero
b. f ( x)  x 6  15x 4  241x 2  225
c. f ( x)  x 4  13x 3  66 x 2  162 x  108 ; 3 - 3i is a zero
5. Step 1 of 4: Use all available methods to factor the polynomial completely. Step 2 of 4: Determine the degree and
enter the y-intercept as an ordered pair. Step 3 of 4: Determine the x-intercept(s) at which f crosses the axis. (First
select the number of x-intercept(s) at which f crosses the axis, then enter the ordered pair(s).) Step 4 of 4: Determine
the zero(s) of f at which it "flattens out". (First select the number of zero(s), then enter the ordered pairs.)
a. f ( x)  x 4  2 x 3  5x 2  4 x  6
b. f ( x)  4 x 3  13x 2  14 x  5