REAL ZEROS OF A POLYNOMIAL FUNCTION

1
SECTION 3.5

• REAL ZEROS OF A POLYNOMIAL FUNCTION

2
THE REAL ZEROS OF A POLYNOMIAL FUNCTION
When we divide one polynomial by another, we
obtain a quotient and a remainder. Thus the
dividend can be written as (Divisor)(Quotient)
Remainder
3
THEOREM DIVISION ALGORITHM FOR POLYNOMIALS
OR f(x) g(x) q(x) r(x)
4
REAL ZEROS OF A POLYNOMIAL FUNCTION
If the divisor is a polynomial of the form x - c
where c is a real number, then the remainder r(x)
is either the zero polynomial or a polynomial of
degree 0. Thus, for such divisors, the remainder
is some number R and we may write f(x) (x -
c) q(x) R
5
REAL ZEROS OF A POLYNOMIAL FUNCTION
If the x variable in the equation of f(x) gets
replaced by the value c, then f(x) (x - c)
q(x) R f(c) (c - c) q(x) R f(c) R
6
REMAINDER THEOREM
Let f be a polynomial function. If f(x) is
divided by x - c, then the remainder is f(c). Ex
Find the remainder if f(x) x3 - 4x2 2x - 5
is divided by (a) x - 3 and (b) x 2
7
FACTOR THEOREM
Let f be a polynomial function. Then x - c is a
factor of f(x) if and only if f(c) 0. Ex Use
the Factor Theorem to determine whether the
function f(x) 2x3 - x2 2x - 3 has the
factor (a) x - 1 and (b) x 3
8
THEOREM NUMBER OF ZEROS
A polynomial function cannot have more zeros than
its degree.
9
DESCARTES RULE OF SIGNS
Let f denote a polynomial function. The number of
positive real zeros of f either equals the
number of variations in sign of the nonzero
coefficients of f(x) or else equals that number
less an even integer.
10
DESCARTES RULE OF SIGNS
Let f denote a polynomial function. The number of
negative real zeros of f either equals the
number of variations in sign of the nonzero
coefficients of f(- x) or else equals that
number less an even integer.
11
EXAMPLE
Discuss the real zeros of f(x) 3x6 - 4x4
3x3 2x2 - x - 3
12
RATIONAL ZEROS THEOREM
Let f be a polynomial function of degree 1 or
higher of the form f(x) a n x n a n-1 x n-1
. . . a1x a0 (an ? 0, a0 ? 0) where each
coefficient is an integer. If p/q, in lowest
terms, is a rational zero of f, then p must be a
factor of a0 and q must be a factor of an.
13
EXAMPLE
f(x) 2x2 - x - 3 (2x - 3)(x 1) zeros
3/2, - 1
14
EXAMPLE
List the potential rational zeros of f(x) 2x3
11x2 - 7x - 6 p ?? 1, ? 2, ? 3, ? 6 q ? 1, ? 2
15
FINDING THE REAL ZEROS OF A POLYNOMIAL FUNCTION
EXAMPLES 5, 6 7
16
THEOREM
Every polynomial function (with real
coefficients) can be uniquely factored into a
product of linear factors and/or irreducible
quadratic factors.
17
COROLLARY
Every polynomial function (with real
coefficients) of odd degree has at least one real
zero.
18
BOUNDS ON ZEROS
We wont worry about this topic.
19
Intermediate Value Theorem
Let f denote a continuous function. If a lt b and
if f (a) and f (b) are of opposite signs, then
the graph of f has at least one x-intercept
between a and b.
20
EXAMPLE
Use the Intermediate Value Theorem to show that
the graph of the function has an x-intercept in
the given interval. Approximate the x-intercept
correct to 2 decimal places. f(x) x4 8x3 - x2
2 - 1, 0
21
f(x) x 4 8x 3 - x 2 2 - 1, 0
f(-1) (- 1)4 8(- 1)3 - (- 1)2 2 1 - 8 -
1 2 - 6 f(0) (0)4 8(0)3 - (0)2 2
0 0 - 0 2 2
22
EXAMPLE
Use the IVT to show that the graph of the
function has an x-intercept in the given
interval. Approximate the x-intercept correct to
2 decimal places. f(x) x 5 - 3x 4 - 2x 3 6x 2
x 2 1.7,1.8 Use your calculator for f(1.7)
and f(1.8).
23

• CONCLUSION OF SECTION 3.5