Real
Zeros:
This section is dedicated to real
zeros and how to find them.
Graphically, a real zero, also known as the root of the
function, is
where the line crosses or touches the x-axis. A real zero
is where the line touches the x-axis at an integer.
Zeros
of Polynomial Functions
A zero of a function is a number x where f(x)=0. (e.g.
f(x)=x-2 and
f(2)=2-2=0 where 2 is the zero). It can be shown that for
a function f of degree n, the following are true:
1.The graph has at most n-1 turning points.
2.The function f has most n real zeros.
If f is a polynomial function and a is a real zero of f,
then the
following statements are equal:
1. x=a is a zero
2. x=a is a solution for f(x)=0
3. (x-a) is a factor of f(x)
4. (a,0) is an x-intercept of f.
Here's an example of how to find zeros:
f(x)=-2x4+2x
=-2x2(x2-1)
=-2x2(x-1)(x+1)
The real zeros are 0, 1, and -1.
Descartes
Rule of Signs
1. The number of positive zeros of
f(x) is either equal to the number
of variations of signs or is less than that number by an
even number.
2.
The number of negative real zeros is either qual to the
number of variations in sign of f(-x) or is less than
that number by an even integer.
Variation in sign means that two consectutive
coefficients have opposite
signs.
e.g. f(x)=3x3-5x2+6x-4
has three variations, whereas
f(-x)=3(-x)3-5(-x)2
+6(-x)-4 has no variations. Thus the Descartes rules of
signs says that f(x) has either 3 or 1 real zeros and no
negative zeros.
Rational Zero Test
The Rational Zero Test relates the possible rational
zeros to the
leading coefficient and the constant term of the
polynomial.
Rational Zero = p/q
p=factor of constant
q=factor of leading coefficient.
You use trial and error to detmermine whether any of the
actual zeros
are actual zeros.
Bounds
Let f(x) be divided by x-c using synthetic division.
1. If c>0 each number in the last row is either
positive or zero then c
is an upper bound for the real zeros of f.
2. If <>0 each number in the last row are
alternately positive and
negative then c is a lower bound for the real zeros of f.
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