Zeros of Polynomial Functions:
A polynomial may be written in COMPLETE FACTORED FORM (p.280) using its zeros and leading
coefficient.
Suppose a polynomial f x   an x n      a2 x 2  a1 x  a0 has n real zeros c1 , c2 , c3 ,  , cn , where distinct zeros
are listed as many times as their multiplicities. Then f x  can be written in complete factored form as
f x   an x  c1 x  c2 x  c3    x  cn .
What about:
5th Degree Polynomial
y
x
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   
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Zeros and Multiplicity –
 If a zero of a polynomial f(x) has odd multiplicity, than its graph crosses the x-axis at the zero.
 If a zero of a polynomial f(x) has even multiplicity, than its graph intersects, but does not cross, the xaxis at the zero.
Examples
f(x) = (x+2)(x+2)(x+2)(x-3)(x-3)
f(x) = (x+2)3(x-3)2
h(x) = (x+3)5(x-5)4
g(x) = -(x+3)(x+3)(x-1)(x-5)
g(x) = -(x+3)2(x-1)(x-5)
y
y
y
x
 

 

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x
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x
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Finding the zeros of a polynomial function:

Sometimes a polynomial function will factor allowing us to find the zeros that way.
o Examples: f(x) = x3 – 5x2 – 6x
g(x) = 2x4 + 11x3 – 21x2

We can use technology (graphing calculator/program) to find zeros from a graph.
o Examples:
f(x) = x4 – 3x3 – 4x2 + 3x + 1
(the zeros are all in the standard viewing window)
g(x) = x5 – 12x4 + 7x3 + 15
(the zeros are not all in the standard viewing window)
How do we pick a good viewing window? Start with the standard window, if you don’t see all of the zeros,
use the following theorem:
Theorem: If r is a zero of a polynomial function P(x) = xn + an-1xn-1+an-2xn-2 + … + a1x + a0, then
r  1  max{ an 1 , an  2 , ..., a1 , a0 } .
To see all of the zeros for the function, first calculate 1  max{ an 1 , an  2 , ..., a1 , a0 } . Then open your
viewing window such that your minimum x is the opposite of the number calculated and your maximum x is
the number calculated.
Examples: Find the zeros of the following polynomials.
y = x5 + 14x4 – 10x2 – 15
y = 4x4 – 40x3 – 123x2 – 29x +50
Application: (Use your calculator’s zero function to solve equations as well.)
The monthly average high temperatures in degrees Fahrenheit at Daytona Beach can be modeled by
f ( x)  0.0151x 4  0.438x3  3.60 x 2  6.49 x  72.5 where x = 1 corresponds to January and x = 12 represents
December.
a) Find the average high temperature during March and July.
b) Estimate when the average high temperature is 80 degrees Fahrenheit.
F(x) is given as 80, so 80  0.0151x 4  0.438 x 3  3.60 x 2  6.49 x  72.5
Get zero on one side, 0  0.0151x 4  0.438 x 3  3.60 x 2  6.49 x  7.5
Graph the function given on the right and find its zeros.
Polynomials can have zeros that are real or imaginary.
Examples: f(x) = x3 + 4x
vs
0 = x(x-2i)(x+2i)

g(x) = x3 – 4x
0 = x(x-2)(x+2)
y
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y
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x
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x
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The Fundamental Theorem of Algebra and the Number of Zeros Theorem (Sec 4.4)
Fundamental Theorem of Algebra
A polynomial f(x) of degree n  1 has at least one complex zero.
Number of Zeros Theorem
A polynomial of degree n has at most n distinct zeros (or exactly n zeros if multiplicity is counted).
Fundamental Theorem of Algebra (alternate form)
Every polynomial of the nth degree with real coefficients has precisely n zeros in C (complex number system).
The fundamental theorem of algebra states that every non-zero single-variable polynomial, with complex
coefficients, has exactly as many complex roots as its degree, if repeated roots are counted up to their
multiplicity. Equivalently, the field of complex numbers is algebraically closed.
In other words, for every complex polynomial p of degree n > 0 the equation p(z) = 0 has exactly n complex
solutions, counting multiplicities.
So the original statement about Complete Factored form becomes:
A polynomial may be written in COMPLETE FACTORED FORM using its zeros and leading
coefficient.
Suppose a polynomial f x   an x n      a2 x 2  a1 x  a0 has n complex zeros c1 , c2 , c3 ,  , cn , where distinct
zeros are listed as many times as their multiplicities. Then f x  can be written in complete factored form as
f x   an x  c1 x  c2 x  c3    x  cn .
Some examples of graphs. Determine the number of real zeros and the number of imaginary zeros.
Degree 2
Degree 4
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y
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Degree 5
Degree 5
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y
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