2.1 Zeros of Polynomial Functions
The Fundamental Theorem of Algebra
- If
is a polynomial of degree n, where n>0, then f has at least one zero in the complex number
system
Study Tip: In order to find the zeros of a function
, set it equal to 0 and solve the resulting equation
Example: Find the zeros of the polynomial
-
Set the equation equal to zero
-
Factor
-
Solving each for zero
-
Complete factorization
The Rational Zero Test
If the polynomial
rational zero of f has the form
has integer coefficients, every
Rational zero =
where p and q have no common factors other than 1, and
p = a factor of the constant
Factors of the constant term
q = a factor of the leading coefficient
Factors of the leading coefficient
EXAMPLE: Find all the rational zeros of the function
- Factors of the constant term are
- Factors of the leading coefficient are
- We can try 1 and use synthetic division and determine that it is a zero
- This leaves us the polynomial
- Factoring this leaves us with
- The rational zeros are therefore
Notice these are all possible
EXAMPLE: Factor
completely.
- Factors are
and
- Possible rational zeros are
- Using the graph we can try -3 and/or 3
- Using synthetic division with 3 we obtain
- Using synthetic division with -3 on the depressed equation we get
- Factoring the last part we get
- This gives is zeros of
Notice that these are all possible
Complex Zeros Occur in Conjugate Pairs
Let
be a polynomial function that has real coefficients. If
function, the conjugate
is also a zero of the function.
, where
, is a zero of the
EXAMPLE: Find a fourth-degree polynomial function with real coefficients that has -1, -1, and 3i as zeros.
-
Conjugate pairs says that -3i is a zero
- Foiling gives us
- Finally we multiply and get
EXAMPLE: Find all the zeros of
given that 3i is a zero.
- Since 3i is a zero, we also have to have -3i as a zero
- Foiling
Conjugate Pairs Theorem
yields
- Use long division and divide
- This results in
which tells us we have zeros at
EXAMPLE: Find all the zeros of
of linear factors.
- List all the possible rational zeros
and write the polynomial as a product
and
giving us
- Using synthetic division we test -2 and get
- Using grouping on the last one yields
- Since we want linear factors, have to factor the last one and we get
giving us zeros at
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