Download polynomial function in x of degree n

Section 2.1 Vocabulary
Definition of Polynomial function
• Let n be a nonnegative integer and
let an,an-1, …, a2, a1, a0 be real
numbers with an not equal to zero.
The function given by
F(x) = anxn + an-1xn-1+….. + a2x2 + a1x + a0
Is called a polynomial function in x of
degree n.
Definition of Quadratic
Function
• Let a, b, and c be real numbers
with a not equal to zero. The
function given by
F(x) = ax2 + bx + c
is called a quadratic function
The graph of a quadratic
function is a special type
of U-shaped curve called
a parabola.
All parabolas are
symmetric with
respect to a line
called the axis of
symmetry.
Standard form of a
Quadratic Function
•The quadratic function
given by
2
F(x) = a(x – h) + k ,
where a cannot equal
zero
Maximum and Minimum values
of Quadratic Functions
• If a > 0, f has a minimum
value at x = -b / (2a)
• If a < 0 , f has a maximum
value at x = -b/(2a)
Section 2.2 Vocabulary
Continuous
• In order to be continuous
the graph of the
polynomial function has no
breaks, holes, or gaps.
Leading Coefficient Test
• If the highest exponent is odd, and the leading
coefficient is positive:
Falls left, rises right
• If the highest exponent is odd, and the leading
coefficient is negative:
Rises left falls right
• If the highest exponent is even, and the leading
coefficient is positive:
Rises left and right
• If the highest exponent is even, and the leading
coefficient is negative:
Falls left and right
A polynomial
n
anx + …+ a1x + a0
has at most n real zeros,
and at most n-1 relative
extrema( minima, or
maxima)
Real Zeros of Polynomial Functions
If a is a real number than the following
statements are equivalent:
• x= a is a real zero of the function
• X = a is a solution of the polynomial equation
f(x) = 0
• (x-a) is a factor of the polynomial f(x).
• (a,0) is an x-intercept of the graph of f(x).
Repeated zeros
• For a polynomial function, a factor of (x – a) k,
k > 1, yields a repeated zero x = a of
multiplicity k.
1. If k is odd, the graph crosses the x-axis at
x = a.
1. If k is even, the graph touches the x-axis at
x=a
Intermediate Value Theorem
• Let a and b be real
numbers such that a < b. If
f is a polynomial function
such that f(a) ≠ f(b), then in
the interval [a,b], f takes on
every value between f(a)
and f(b).
Section 2.3 Vocabulary
2 methods of
dividing polynomials
•Long Division
•Synthetic Division(only
when the divisor has the
form x – k)
Division algorithm
• If f(x) and d(x) are polynimials where d(x)
≠ 0, and the degree of d(x) is less than or
equal to the degree of f(x), there exists
unique polynomials q(x) and r(x) such
that:
f(x) = d(x) q(x) + r(x)
Note: f(x) / d(x) is improper
r(x) / d(x) is proper
The remainder theorem
•If a polynomial f(x) is
divided by x – k, the
remainder is r = f(k).
The Factor Theorem
•A polynomial f(x)
has a factor of
(x – k) if and only if
f(k) = 0
Descartes' Rule of Signs
• Let f(x) = anxn + …+ a1x + a0 be a polynomial with
real coefficients and a0 ≠ 0.
1. The number of positive real zeros of f is either
equal to the number of variations of the sign of
f(x) or less than that number by an even integer
2. The number of negative real zeros of f is wither
equal to the number of variations of the sign of
f(-x) or less than that number by an even integer
A variation in sign
means that two
consecutive (non zero)
coefficients have
opposite signs.
A real number b is an upper bound for
the real zeros of f is no zeros are
greater than b.
Similarly, b is a lower bound if no real
zeros of f are less than b.
Section 2.4 Vocabulary
Complex number
Has a real part(a) and
an imaginary part (bi)
and is written in
standard form:
a + bi
Equality of complex numbers
• Two complex numbers a + bi
and c + di, written in standard
form, are equal to each other
a + bi = c + di
If and only if a = c and b = d
Pairs of complex
numbers of the
forms a + bi and a –
bi are called complex
conjugates
Section 2.5 Vocabulary
The Fundamental Theorem of
Algebra
If f(x) is a polynomial of
degree n, where n > 0, then
f has at least one zero in
the complex number
system.
Linear Factorization Theorem
If f(x) is a polynomial of degree n,
there n > 0, f has precisely n
linear factors
f(x) = an(x – c1) (x – c2) …..( x – cn)
Where c1, c2,…., cn are complex
numbers.
A quadratic factor with
no real zeros is said to be
prime or irreducible over
the reals.
Section 2.6 Vocabulary
A rational function
can be written in
the form:
f(x) = N(x) / D(x)
The line x = a is a vertical
asymptote of the graph
of f if f(x) approaches
infinity or f(x) approaches
negative infinity as x
approaches a, either from
the right or from the left.
The line y = b is a
horizontal asymptote of
the graph of f if f(x)
approaches b as x
approaches positive or
negative infinity.
Section 2.7 Vocabulary
Consider a rational function.
If the degree of the
numerator is exactly one
more than the degree of the
denominator then the
function has a slant
asymptote.