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) =
If a polynomial f (x) is divided by (x − c), then the
remainder r = f (c).
Example: Find the remainder if f (x) = x^4 − 6x^3 +
divided by (x + 2). Use
the Remainder Theorem
Example: Use the Remainder Theorem to find f (3) if
Note: If the polynomial f (x) is divided by (x − c)
the remainder r = 0, then f (c) = 0, that is, c is a
zero of f (x).
Example: If f (x) = x^3 − 6x + 4, is 2 a zero of f
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 x^3 − 2x +1?
Example: Factor f (x) into linear factors given that c is
zero of f (x).
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
zeros if to count each zero as many times as its
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:
Example: Find a polynomial of the lowest degree
possible with only real coefficients which has the given
Bounds of Zeros
Let M > 0.
A number M is called a bound on zeros of a polynomial f
if every real zero c lies between – M and M, inclusive,
−M ≤ c ≤ M
Theorem: Let f denote a polynomial of degree n
leading coefficient is 1.
A bound M on the real zeros of f is the smallest of
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
(b) Use a graphing utility to graph the polynomial in the
viewing rectangle de termined 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.
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