College Algebra Lecture Notes
Section 3.3
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Section 3.3: The Zeroes of Polynomial Functions
Big Idea: Since all algebraic equations can be converted to a polynomial equation where a single
polynomial function is equal to zero, finding the zeroes of a polynomial function is the most
important skill in algebra.
Big Skill: .You should be able to find the zeroes of a polynomial function using the techniques
below.
A. The Fundamental Theorem of Algebra
Complex Polynomial Functions
A complex polynomial of degree n has the form
P  x   an xn  an1 xn1   a1 x1  a0
Where an , an1 ,
, a0 are complex numbers and an  0 .
The Fundamental Theorem of Algebra
Every complex polynomial of degree n  1has at least one complex zero.
The Linear Factorization Theorem
If p  x  is a polynomial of degree n  1, then p has exactly n linear factors and can be written in
the form:
p  x   a  x  c1  x  c2   x  cn 
where a  0 and c1 , c2 ,
, cn are (not necessarily distinct) complex numbers.
Zeroes of Multiplicity
If p  x  is a polynomial of degree n  1, and  x  c  occurs as a factor of p exactly m times, then
c is a zero of multiplicity m.
Corollary I: Irreducible Quadratic Factors
If p  x  is a polynomial with real coefficients, then p can be factored into a product of linear
factors (which are not necessarily distinct) and irreducible quadratic factors, all of which have
real coefficients.
Corollary II: Complex Conjugates
If p  x  is a polynomial with real coefficients, then complex zeros must occur in conjugate pairs.
That is, if a  bi, b  0 is a zero, then a  bi will also be a zero.
Corollary III: Number of Zeroes
If p  x  is a polynomial of degree n  1, the p has exactly n zeroes (either real or complex),
where zeroes of multiplicity m are counted m times.
College Algebra Lecture Notes
Section 3.3
Practice:
1. Find the linear factors and all zeroes of p  x   x4  21x2  100 .
2. Find the linear factors and all zeroes of p  x   2x3  4x2  2x .
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College Algebra Lecture Notes
Section 3.3
3. Find the linear factors and all zeroes of p  x   x3  4 x2  16 x  64 .
4. Find the linear factors and all zeroes of p  x   x3  x .
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College Algebra Lecture Notes
Section 3.3
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B. Real Polynomials and the Intermediate Value Theorem
 Note that the theorems on the previous pages only state the fact that solutions do exist;
they say nothing about how to find those solutions.
 In the following pages, we look at theorems that help us find actual values for the
solutions.
 There are several different theorems for several different scenarios.
 The Intermediate Value Theorem allows us to verify that a solution exists somewhere
within an interval of numbers, and can also be used to zoom in closer on a value.
The Intermediate Value Theorem
If P  x  is a polynomial with real coefficients, and P  a  and P  b  have opposite signs, then
there is at least one value c between a and b such that P  c   0 .
Practice:
5. Use the intermediate value theorem to verify that P  x   x3  2x  0.5 has a zero in the
interval x 1, 2 .
6. Use the intermediate value theorem to narrow down the interval in which that zero of
P  x   x3  2x  0.5 occurs.
College Algebra Lecture Notes
Section 3.3
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C. The Rational Zeroes Theorem
The Rational Zeroes Theorem
If P  x  is a polynomial with integer coefficients, then any existing rational zeros of P  x  must
be of the form of a rational number
of the leading coefficient.

p
q
, where p is a factor of the constant term and q is a factor
Notice that if we have the special case of a polynomial with integer coefficients, then the
rational zeroes theorem predicts some possible answers for us, and all we need to do is
verify which of those predictions are true zeroes.
Tests to Determine if 1 or -1 is a Zero of a Polynomial
If P  x  is a polynomial with real coefficients, then


If the sum of all coefficients is zero, then 1 is a root and (x – 1) is a factor.
If after changing all the signs of all terms with odd degree the sum of all coefficients is
zero, then -1 is a root and (x + 1) is a factor.

Notice that this theorem gives us a shortcut for checking two of the always possible zeros
of a polynomial with integer coefficients.
Practice:
7. Find the zeroes of the polynomial function p  x    2x 1 3x  2 5x  7  , then expand
the polynomial, list all possible zeroes predicted by the rational zeroes theorem, and note
that the actual zeroes are in that set.
College Algebra Lecture Notes
Section 3.3
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8. Find the factors and zeroes of the polynomial function p  x   x3  21x  20 using the
rational zeroes theorem.
9. Find the factors and zeroes of the polynomial function p  x   9 x3  7 x  2 using the
rational zeroes theorem.
College Algebra Lecture Notes
Section 3.3
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D. Descartes’ Rule of Signs and Upper/Lower Bounds
Descartes’ Rule of Signs
If P  x  is a polynomial with real coefficients, and we want to solve P  x   0 ,

The number of positive real zeroes is equal to the number of variations in the signs of the
coefficients of P  x  , or an even number less.

The number of negative real zeroes is equal to the number of variations in the signs of the
coefficients of P   x  , or an even number less.

Notice that this theorem can be used to make factoring more efficient, because if we first
test for the number of positive and negative roots, then we know we can stop checking
when we find that number of each kind of root.
Upper and Lower Bounds Property
If P  x  is a polynomial with real coefficients, then

If P  x  is divided by  x  b  , b  0 using synthetic division and all coefficients in the
quotient row are either positive or zero, then b is an upper bound on the zeros of P  x  .

If P  x  is divided by  x  a  , a  0 using synthetic division and all coefficients in the
quotient row alternate in sign, then a is a lower bound on the zeros of P  x  .

Zero coefficients in the quotient row can be either positive or negative as needed.

Notice that this theorem also makes factoring more efficient, because if we start with
small factors and work our way out until the properties apply, then we know we can stop.
Practice:
10. Construct a polynomial with three positive rational roots and notice how the signs of the
coefficients alternate. Also list all possible rational roots, and notice how the upper and
lower bounds properties apply.
College Algebra Lecture Notes
Section 3.3
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11. Construct a polynomial with one positive root and a pair of complex conjugate roots.
Notice how the signs of the coefficients still alternate, but that there is only 3 – 2 = 1
positive root.
12. Construct a polynomial with three negative roots and notice how the signs of the
coefficients of P(-x) alternate. Also list all possible rational roots, and notice how the
upper and lower bounds properties apply.
College Algebra Lecture Notes
Section 3.3
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13. Construct a polynomial with one negative root and a pair of complex conjugate roots.
Notice how the signs of the coefficients of P(-x) still alternate, but that there is only 3 – 2
= 1 negative root.
14. Find the factors and zeros of g  x   3x4  4 x3  21x2 10 x  24 .
College Algebra Lecture Notes
E. Applications of Polynomial Functions
Practice:
15. .
Section 3.3
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