Document Sample

```					        SECTION 3.5

REAL ZEROS OF A POLYNOMIAL
FUNCTION
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
THEOREM: DIVISION
ALGORITHM FOR
POLYNOMIALS

f(x)          r(x)
 q(x) 
g(x)          g(x)

OR
f(x) = g(x) q(x) + r(x)

Dividend divisor quotient remainder
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
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
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
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
THEOREM: NUMBER OF
ZEROS

A polynomial function cannot have
more zeros than its degree.
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.
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.
EXAMPLE

Discuss the real zeros of
f(x) = 3x6 - 4x4 + 3x3 + 2x2 - x - 3
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.
EXAMPLE

f(x) = 2x2 - x - 3
(2x - 3)(x + 1)
zeros: 3/2, - 1
EXAMPLE

List the potential rational zeros of
f(x) = 2x3 + 11x2 - 7x - 6
p:  1,  2,  3,  6
q:  1,  2
FINDING THE REAL ZEROS OF
A POLYNOMIAL FUNCTION

EXAMPLES 5, 6 & 7
THEOREM

Every polynomial function (with real
coefficients) can be uniquely factored
into a product of linear factors and/or
COROLLARY

Every polynomial function (with real
coefficients) of odd degree has at least
one real zero.
BOUNDS ON ZEROS

Intermediate Value
Theorem
Let f denote a continuous function.
If a < 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.
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]
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
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).
CONCLUSION OF SECTION 3.5

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 5 posted: 5/30/2012 language: English pages: 23
How are you planning on using Docstoc?