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A polynomial function of degree n is
where the a’s are real numbers and the n’s are nonnegative integers
and an 0.
Polynomial Function
01
1)( axaxaxf nn
nn
A polynomial function of degree 2 is called a quadratic function.
It is of the form
a, b, and c are real numbers and a 0.
Quadratic Function
cbxaxxf 2)(
For a quadratic function of the form
gives the axis of
symmetry.
Axis of Symmetry
cbxaxxf 2)(
a
bx
2
A quadratic function of the form
is in standard form.
axis of symmetry: x = hvertex: (h, k)
Standard Form
0,)()( 2 akhxaxf
Characteristics of Parabola
symmetryofaxis
symmetryofaxisa > 0
a < 0
vertex: minimum
vertex: maximum
The graph of a polynomial function…
1. Is continuous.
2. Has smooth, rounded turns.
3. For n even, both sides go same way.
4. For n odd, sides go opposite way.
5. For a > 0, right side goes up.
6. For a < 0, right side goes down.
Characteristics
.
xas
xf )(
xas
xf )(
an < 0
xas
xf )(
xas
xf )(
graphs of a polynomial function for n odd:0
11)( axaxaxf n
nn
n
Leading Coefficient Test: n odd
an > 0
.
an < 0
graphs of a polynomial function for n even:0
11)( axaxaxf n
nn
n
an > 0
xas
xf )(
xas
xf )(
xas
xf )(
xas
xf )(
Leading Coefficient Test: n even
The following statements are equivalent for
real number a and polynomial function f :
1. x = a is root or zero of f.
2. x = a is solution of f (x) = 0.
3. (x - a) is factor of f (x).
4. (a, 0) is x-intercept of graph of f (x).
Roots, Zeros, Solutions
1. If a polynomial function contains a factor (x - a)k, then x = a is a repeated root of multiplicity k.
2. If k is even, the graph touches (not crosses) the x-axis at x = a.
3. If k is odd, the graph crosses the x-axis at x = a.
Repeated Roots (Zeros)
If a < b are two real numbers
and f (x)is a polynomial function
with f (a) f (b),
then f (x) takes on every real
number value between
f (a) and f (b) for a x b.
Intermediate Value Theorem
Let f (x) be a polynomial function and a < b be two real numbers.
If f (a) and f (b)
have opposite signs
(one positive and one negative),
then f (x) = 0 for a < x < b.
NOTE to Intermediate Value
If f (x) and d(x) are polynomialswith d(x) 0 and the degree of d(x) isless than or equal to the degree of f(x),
then q(x) and r (x) are uniquepolynomials such that
f (x) = d(x) ·q(x) + r (x)where r (x) = 0 or
has a degree less than d(x).
Full Division Algorithm
f (x) = d(x) ·q(x) + r (x)
dividend quotient divisor remainder
where r (x) = 0 orhas a degree less than d(x).
Short Division Algorithm
ax3 + bx2 + cx + d divided by x - k
k a b c d
ka
a r coefficients of quotient remainder
1. Copy leading coefficient.
2. Multiply diagonally. 3. Add vertically.
Synthetic Division
a’s are real numbers, an 0, and a0 0.
1. Number of positive real zeros of f equals number of variations in sign of f(x), or less than that number by an even integer.
2. Number of negative real zeros of f equals number of variations in sign of f(-x), or less than that number by an even integer.
Descartes’s Rule of Signs
01
1)( axaxaxf nn
nn
a’s are real numbers, an 0, and a0 0.
1. f(x) has two change-of-signs; thus, f(x) has two or zero positive real roots.
2. f(-x) = -4x3 5x2 + 6 has one change-of-signs; thus, f(x) has one negative real root.
Example 1: Descartes’s Rule of Signs
654)( 23 xxxf
Factor out x; f(x) = x(4x2 5x + 6) = xg(x)
1. g(x) has two change-of-signs; thus, g(x) has two or zero positive real roots.
2. g(-x) = 4x2 + 5x + 6 has zero change-of-signs; thus, g(x) has no negative real root.
Example 2: Descartes’s Rule of Signs
xxxxf 654)( 23
If a’s are integers, every rational zero of f has the form
rational zero = p/q,
in reduced form, and p and q are factors of a0 and an, respectively.
Rational Zero Test0
11)( axaxaxf n
nn
n
f(x) = 4x3 5x2 + 6
p {1, 2, 3, 6}
q {1, 2, 4}
p/q {1, 2, 3, 6, 1/2, 1/4, 3/2, 3/4}
represents all possible rational roots of f(x) = 4x3 5x2 + 6 .
Example 3: Rational Zero Test
f(x) is a polynomial with real coefficients and an > 0 with
f(x) (x - c), using synthetic division:
1. If c > 0 and each # in last row is either positive or zero, c is an upper bound.
2. If c < 0 and the #’s in the last row alternate positive and negative, c is an lower bound.
Upper and Lower Bound
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