English | Español

Try our Free Online Math Solver!

Online Math Solver












Please use this form if you would like
to have this math solver on your website,
free of charge.

Zeros of a Polynomial Function

L19 Zeros of a Polynomial Function

Division Algorithm for Polynomials

If f and g are two polynomials and g is not the
zero polynomial, then there are the unique
polynomials q (quotient) and r (remainder) such

where r(x) either the zero polynomial or of degree
less than the degree of g(x).

Note: If g(x) = x − c, then f (x) = q(x)(x − c) + r ,
where r is a number.

If x = c, then f (c) =

Remainder Theorem

If a polynomial f (x) is divided by (x − c), then the
remainder r = f (c).

Example: Find the remainder if f (x) = x4 − 6x3 + 2 is
divided by (x + 2). Use
synthetic division

the Remainder Theorem

Example: Use the Remainder Theorem to find f (3) if
f (x) = x6 − 6x5 + 54x2 −16x +1

Note: If the polynomial f (x) is divided by (x − c) and
the remainder r = 0, then f (c) = 0, that is, c is a
zero of f (x).

Example: If f (x) = x3 − 6x + 4, is 2 a zero of f ?

Factor Theorem

The polynomial (x − c) is a factor of the polynomial
f (x) if and only if f (c) = 0.

If (x − c) is a factor of f , then f (x) =
hence f (c) =

If f (c) = 0, then r = f (c) = and f (x) =
therefore (x − c) is a factor of f .

Example: Is (x −1) a factor of x3 − 2x +1?

Example: Factor f (x) into linear factors given that c is a
zero of f (x).

f (x) = 3x3 − 5x2 −16x +12;
c = −2

Fundamental Theorem of Algebra

Every polynomial of degree 1 or more has at least
one complex zero .

Let deg f (x) = n, n ≥1

Number of Zeros Theorem

A polynomial of degree n has at most n distinct zeros.

Note: A polynomial of degree n has exactly n complex
zeros if to count each zero as many times as its
multiplicity .

Conjugate Zeros Theorem

If f (x) is a polynomial whose coefficients are real ,
and if a + bi is a zero of f (x), with a and b real
numbers, then a − bi is also a zero of f (x).

Example: One zero is given, find all others

Example: Find the polynomial of degree 3 with real
coefficients that satisfies the conditions:

zeros @ −2,1, 0

 f (−1) = −1

Example: Find a polynomial of the lowest degree
possible with only real coefficients which has the given

6 − 3i

Bounds of Zeros

Let M > 0.
A number M is called a bound on zeros of a polynomial f
if everyreal zero c lies between – M and M, inclusive,
that is,

−M ≤ c ≤ M

Theorem: Let f denote a polynomial of degree n whose
leading coefficient is 1.

A bound M on the real zeros of f is the smallest of the
two numbers:

Notes: 1) If the leading coefficient of the polynomial
f (x) is not 1, that is, an≠1 , in order to use the Theorem
you can replace f (x) with the polynomial
since f ( x) has the same zeros as g(x).

2) The bounds on the zeros give a good choice of
Xmin and Xmax of the viewing rectangle since all the x- intercepts
of the graph can be seen .

Example: (a) Find bounds on the real zeros of the
polynomial function f ( x) = −3x3 + 8x2 − 6x + 9.

(b) Use a graphing utility to graph the polynomial in the
viewing rectangle determined by the bounds on the zeros.
Approximate the real zeros (x-intercepts).

Intermediate Value Theorem

If f (x) is a polynomial with only real coefficients
and if for real numbers a and b, the values f (a) and
f (b) are of opposite signs , then there exists at least
one real zero between a and b.

Example: Show that the polynomial has a real zero
between 2 and 3.
f (x) = 2x3 − 9x2 + x + 20

Prev Next