Dividing Polynomials (2.3)
Use long division to divide a polynomial
by a divisor by:
(1) dividing the first term of by the
first term of and writing it as a
term of the quotient
(2) multiplying the term of by every
term of
(3) subtracting the product from
(4) bringing down the next term of
(5) repeating the previous steps until the
remainder cannot be divided
A polynomial function can be written as:
Use synthetic division to divide a polynomial
by a divisor by:
(1) writing a in a box and the coefficients
of in a row to the right
(2) writing the leading coefficient of
in the bottom row
(3) multiplying a times the value on the
bottom row and writing it in the middle
row of the next column
(4) adding the top and middle row of the
next column and writing the sum in
the bottom row
(5) repeating steps 3 and 4 until all
columns are filled
(6) writing the quotient and remainder
using descending powers of x
If is divided by , then the
remainder is . If , then
is a factor of .
If
has integer coefficients, then all rational
zeros must be of the form , where p
is a factor of a0 and q is a factor of an.
Zeros of a Polynomial Function (2.4)
The number of positive real zeros of
is the number of sign changes of , or
is less than that by an even number.
The number of negative real zeros of
is the number of sign changes of , or
is less than that by an even number.
If is divided by and the numbers
in the bottom row are all nonnegative, then
k is an upper bound on the zeros of .
If is divided by and the numbers
in the bottom row alternate in sign, then k
is a lower bound on the zeros of .
Partial Fractions (7.5)
Decompose a rational expression
into partial fractions by:
(1) writing the expression as the
sum of fractions with unknown
constants in the numerators
(2) multiplying the resulting
equation by the LCD
(3) equating coefficients of all
powers of x
(4) solving the resulting system
of linear equations
Alternately, steps 3 and 4 can be
replaced by:
(3) repeatedly substituting values
of x into the equation to solve
for each constant one at a time
Sequences (9.1)
A sequence is a function f with a
domain of all natural numbers.
The values are
called the terms of the sequence.
For the sequence , the
partial sums are given by:
The partial sum of a sequence Sn is
written in sigma notation as:
.
The infinite series of a sequence is
written as:
Arithmetic Sequences (9.2)
An arithmetic sequence has the form
. The first term
is a1, and the common difference is d.
The nth term of an arithmetic sequence
is given by:
The nth partial sum of an arithmetic
sequence is given by:
or
Geometric Sequences (9.3)
A geometric sequence has the form
. The first term is a1,
and the common ratio is r.
The nth term of a geometric sequence
is given by:
The nth partial sum of a geometric
sequence is given by:
The sum of an infinite geometric
series is given by:
Pascal’s Triangle (9.5)
In the triangular pattern of coefficients in the
binomial expansions of for increasing
values of n, each coefficient is equal to the sum
of the two coefficients above it.
Binomial Theorem (9.5)
The binomial expansion of is given by:
, where .
Quotient Identities (4.3)
Reciprocal Identities (4.3)
Pythagorean Identities (4.3)
Even/Odd Identities (4.3)
Cofunction Identities (5.3)
Function Values for Common Arcs (4.3)
Θ / 0 / 30 / 45 / 60 / 90 / 120 / 135 / 150 / 180 / 210 / 225 / 240 / 270 / 300 / 315 / 330 / 360T / 0
sin / 0 / 1 / 0 / -1 / 0
cos / 1 / 0 / -1 / 0 / 1
tan / 0 / 1 / U / -1 / 0 / 1 / U / -1 / 0
cot / U / 1 / 0 / -1 / U / 1 / 0 / -1 / U
sec / 1 / 2 / U / -2 / -1 / -2 / U / 2 / 1
csc / U / 2 / 1 / 2 / U / -2 / -1 / -2 / U
Trigonometric Equations (5.2)
When solving trig equations,
(1) simplify the equation so that only
one trig function appears
(2) use algebraic techniques to solve
for the trig function
(3) use inverse trig functions to solve
for the angle
(4) check the solution(s)
Sum and Difference Formulas (5.3)
Double Angle Formulas (5.4)
Power-Reducing Formulas (5.4)
Half-Angle Formulas (5.4)
Special Angle Formulas (5.3, 5.4)
Trig Function / Big + / Small + / Big – / Small –Product-to-Sum Formulas (5.5)
Sum-to-Product Formulas (5.5)
Vectors (6.4)
A scalar is a quantity that specifies
size only.
A vector is a quantity that specifies
both a size and a direction.
The vector with initial point P and
terminal point Q is written .
The magnitude (or size) of a vector
is written and is a scalar quantity.
The position vector has an
initial point and terminal point .
A vector from to can be
written as a position vector .
Vector Arithmetic (6.4)
(1) (2)
(3) (4)
Unit Vectors (6.4)
The unit vector has magnitude 1.
The standard unit vectors are
and .
A vector is written using its standard
components as .
The direction angle θ is given by
and .
A vector is written using its magnitude
and direction as .
The Dot Product (6.5)
The dot product of two vectors is the
scalar .
The angle between two vectors is given
by .
Two vectors are parallel if .
Two vectors are orthogonal if .
Properties of the Dot Product (6.5)
Vector Projection and Decomposition (6.5)
The vector projection of v onto w is
.
The scalar projection of v onto w is
.
The decomposition of w with respect to v is
and .
The work done by a force F moving an object
from P to Q is
.
Polar Coordinates (6.6)
The polar coordinates and
the rectangular coordinates
are related as follows:
Trigonometry and Complex Numbers (6.7)
Def’n The trigonometric form of the
complex number is
given by: ,
where , ,
, and .
Complex Products and Quotients (6.7)
Rule If and
, then
and
.
Complex Powers and Roots (6.7)
Rule If , then
.
Rule If , then the nth
roots of z are given by:
for .
Parabolas (8.2)
A parabola is the set of all points in a
plane equidistant from a fixed point
(focus) and a fixed line (directrix).
The graph of:
is a parabola with vertex , focus
, directrix , and
focal diameter 4a.
The graph of:
is a parabola with vertex , focus
, directrix , and
focal diameter 4a.
Ellipses (8.3)
An ellipse is the set of all points in a
plane the sum of whose distances from
two fixed points (foci) is a constant.
The graph of:
is an ellipse with vertices ,
minor radius b, foci , where
, focal radius , and
eccentricity .
The graph of the equation
is an ellipse with vertices ,
minor radius b, foci , where
, focal radius , and
eccentricity .
Hyperbolas (8.4)
A hyperbola is the set of all points in a
plane the difference of whose distances
from two fixed points (foci) is a constant.
The graph of:
is a hyperbola with vertices ,
asymptotes , foci ,
where , focal radius ,
and eccentricity .
The graph of:
is a hyperbola with vertices ,
asymptotes , foci ,
where , focal radius ,
and eccentricity .
Eccentricity (8.6)
The eccentricity e of a conic is
given as:
If , the conic is a parabola.
If , the conic is an ellipse.
If , the conic is a hyperbola.
Polar Equations of Conics (8.6)
The polar equation of a conic with
one focus at the pole, a directrix
p units from the pole, and an
eccentricity e is given by:
Parametric Equations (8.7)
The equations and
are parametric equations for a plane
curve with parameter t.
Find an equation for a plane curve in
rectangular coordinates by eliminating
the parameter.