11-3 The Short-Run Behavior Polynomial Functions and Zeros
Honors Precalculus
~NOTES~ Day 2
When a polynomial function is presented in standard form (not factored) it is difficult to determine the zeros,
therefore we need to know how to factor a polynomial. In this section we will explore methods to find the
linear factors of a polynomial. Sometimes these factors are impossible to find over the real number set, and
some polynomials are impossible to factor at all! We will only be concerned with simple polynomials for our
purposes in this class.
Finding Zeros of a Polynomial.
If P is a polynomial function, then z is called a zero of P if P  z   0 . In other words, the zeros of P are
the solutions of the polynomial equation P  x   0 . Note that if P  z   0 , and z is a real number, then the
graph of P has an x-intercept at x  z , so the x-intercepts of the graph are zeros of the function.
The Fundamental Theorem of Algebra (a few different forms )

If P  x  is a polynomial of degree n then P  x  will have exactly n zeros, some of which may repeat.

Every nonzero polynomial P  x  with coefficients in
(complex number set) can be factored, in
essentially a unique way, as a product of a constant and linear terms, in the form
P  x   a  x  z1    x  zn  where a is the leading coefficient of P  x  and n is the degree of P  x  .

Every polynomial can be factored (over the real numbers) into a product of linear factors and
irreducible quadratic factors.
FYI Note: Another name for a zero is a “root”.
The Rational Root Theorem
p
If the rational number x  is a zero of the nth degree polynomial, P  x   an x n    a0 where all
q
coefficients are integers then p will be a factor of a0 and q will be a factor of an .
When you apply the rational root theorem, you find all the rational roots, if there are any. If the theorem finds no roots,
the polynomial has no rational roots. (For a cubic, we would observe that the polynomial is irreducible over the rationals.
This is because a factorization of the cubic is either the product of a linear factor and a quadratic factor or it is the product
of three linear factors. Since in either case there is a linear factor, there would be a root in the rationals. With no rational
root, we're done.)
Determining whether a polynomial has rational roots is done by means of the rational root test. There is no prior test one
uses to determine whether to use the rational roots test; one starts with the test to find and remove all the "easy" factors.
Only after all the easy roots are removed (or not found) do you move on to hard tools like the cubic formula, which we
will not be learning at this level 
Using the Rational Root Theorem to find rational zeros
1. Arrange the polynomial in descending order
2. Write down all the factors of the constant term a0 . These are all the possible values of p .
3. Write down all the factors of the leading coefficient an . These are all the possible values of q .
4. Write down all the possible values of
p
p
p
. Remember that since factors can be negative,
and 
q
q
q
must both be included. Simplify each value and cross out any duplicates.
5. Use synthetic division (or long division) to determine the values of
 p
p
for which P    0 . These
q
q
are all the rational roots (zeros) of P(x) .
Find the factored form and the zeros the Polynomial Function.
Ex 1: Given the polynomial function P  x   x3  2 x2  5x  6 :
a) Use the rational root theorem to write the polynomial in factored form.
b) Use the factored form of P  x  to solve the corresponding polynomial equation P  x   0
Ex 2: Given the polynomial function P  x   12 x3  41x 2  38x  40 :
a) Use the rational root theorem to write the polynomial in factored form.
b) Use the factored form of P  x  to solve the corresponding polynomial equation P  x   0
The Irrational Conjugate Roots Theorem
Let P  x  be any polynomial with rational coefficients. If a  b c is a zero of P  x  , where
c is irrational
and a and b are rational, then another root is a  b c .
Ex 3: Given the polynomial function P  x   x4  x3  5x2  3x  6 :
a) Use the rational root theorem to write the polynomial in factored form.
b) Use the factored form of P  x  to solve the corresponding polynomial equation P  x   0
Complex Conjugate Root Theorem
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real
coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root
of P.
Ex 3: Given the polynomial function P  x   x5  5x4  5x3  x2  6 x  16 :
a) Use the rational root theorem to write the polynomial in factored form.
b) Use the factored form of P  x  to solve the corresponding polynomial equation P  x   0
11-3 Day 2 Homework Problems
Please do your homework on a separate paper  Show all work.
Use the Rational Root Theorem and synthetic division to find the zeros of each polynomial function.
1.
f  x   2 x3 11x2  2 x  15
2. P  x   x3  2 x2  9 x  18
3. P  x   3x3 11x2  6 x  8
4. P  x   25x4  40x3 19x2  2x
5. P  x   3x4  5x3  81x  135
6. f  x   6 x4  23x3  51x2  158x  120
7. g  x   2 x5  21x4  43x3  86 x2  348x  280
8. f  x   x4  x3  9 x2  3x  36
9. x5  x4  15x3  21x2  16x  20  P  x 
10. P  x   2 x4  x3  3x2  3x  9