Download Unit 1-Polynomials

Binomial Theorem- Expand and find the ______- term
Unit 1-Polynomials
Rational Expressions
1. Simplify Rational Expressions
a. Factor Numerator and Denominator (look for GCF’s first)
b. Find Excluded Values
c. Divide Out (cancel out) Common Factors
1. Set up Pascal’s Triangle to find the coefficients for each term
2. Count down exponents from first term until there are none left
3. Skip first term, and then start with an exponent of 1 and count up on
the second term.
4. Each pair of exponents should add up to equal the same amount.
Remainder Theorem
2. Multiply Rational Expressions
a. Factor Numerators and Denominators (look for GCF’s first)
b. Find Excluded Values
c. Divide Out (cancel out) Common Factors
d. Simplify by multiplying across
3. Divide Rational Expressions
a. Factor Numerators and Denominators (look for GCF’s first)
b. Keep Change Flip (multiply by reciprocal)
c. Find Excluded Values (don’t forget original denominator)
d. Divide Out (cancel out) Common Factors
e. Simplify by multiplying across
4. Add/Subtract Rational Expressions
a. Factor Denominators
b. Find Least Common Denominators (LCD)
c. Combine
5. Asymptotes and Holes
a. Horizontal Asymptotes
i.
no horizontal asymptote
ii.
iii.
horizontal asymptote at y = 1
divide the leading coefficients to
find where the
horizontal asymptote will be
b. Points of Discontinuity- what x values will give you zero in the
denominator
i. Vertical asymptote- factor won’t cancel
ii. Holes- factor will cancel
1. Substituting an x-value into the polynomial will give a y-value that
matches the remainder if you were to use synthetic division or long
division
Synthetic and Long Division
1. Use the method you feel most comfortable with
Graphing Polynomials
1. General Shape
a. Determined by Degree (largest exponent)
b. End Behavior Determined by leading coefficient (+ or -)
c. Maximum Number of Turning Points- degree minus 1
2. x-intercepts give you roots, solutions, or zeroes (different names for
same thing)
3. y-intercepts give you starting point or initial value in word problems
4. Domain- x-values (input)
a. Usually all real numbers (anything can go in for x: positives,
negatives, or zero)
b. Sometimes restricted:
i. Square Roots- can’t have a negative under radical
ii. Fractions- can’t divide by zero
5. Range- y-values (output)
a. Based on what values we can use for x.
Additional Information:
Unit 2- Solutions to Polynomials and Conics
Factoring
1. Use the X-factor
a. × on the top
b. on the bottom
c. Factors go along the sides; multiply to make the top and add
to make the bottom
d. Divide these factors by your a. Simplify
e. Rewrite as “bottom times x ± top” as the factors
2. When set equal to zero and solved for, will give you solutions (zeroes,
x-intercepts, roots)
Complete the Square- (if you forget this, use max/min on calculator)
1. Divide everything by a if it is something other than 1.
2. Take your b, divide it by 2 and square it. This will give you what c
needs to be to make a perfect square trinomial.
3. Factor as a perfect square trinomial.
4. This is how to write a parabola in vertex form.
Conics- use complete the square if not in that form
1. Parabolas
a. You can be given any two of the following: vertex (point ℎ, on parabola), focus (point inside parabola), and directrix (the
line as x = or y =)
b. Do a quick sketch to see whether it is an x = or y = parabola.
= − ℎ + or = − + ℎ
c. Find your p value (distance between vertex and
focus/directrix)
d. Plug values into the above equation.
2. Circles
a. In the form − ℎ + − = where ℎ, is the
center and r is the radius
Fundamental Theorem of Algebra
1. Your degree will tell you how many total solutions you have for any
given polynomial.
2. Some of the solutions can be:
a. Imaginary (do not cross the x-axis)b. Found by using the quadratic formula:
=
±√ !
c. Repeated (makes the graph “bounce” off of the x-axis)
Additional Information:
Unit 3- Standard Deviation and Normal Distribution
Normal Distribution
1. Enter values into calculator:
a. STAT
b. EDIT
c. Data goes here
d. STAT
e. C ALC
f. 1: 1-Var Stats
2. Mean is ; Standard Deviation is σx; Median is Med
Finding Percent Given the Lowest, Highest, Mean, and Standard Deviation
Calculator Steps
1.
2.
3.
4.
5.
2nd VARS
2: normalcdf (
Enter in order: lowest, highest, mean, standard deviation
ENTER
Multiply your answer by 100 to find the percent OR multiply the
answer by the total to find out about how many values were within
that range.
Taking Samples
1. Know the differences between the following:
a. Samples (simple random, systematic random, convenience)
b. Observational Study vs. Experiment
Additional Information:
Unit 4- Functions
1. Transformation of functions
a. Any movement within the grouping symbols by the x
(parenthesis, square roots, absolute value, etc…) moves the
function left (when +) and right (when -). Sometimes it is
written as " + b. Any movement outside on the end of the equation moves the
function up (when +) or down (when -) Sometimes it is written
as " + c. Anything out in front will vertically stretch (when > 1) or
compress (when between 0 and 1). When this value is
negative, it flips the function over the x-axis. Sometimes it is
written as "
2. Inverse Functioins
a. Switch your x and y
b. Solve the equation for y using inverse operations. When an
exponential, you will use log and vice versa.
Additional Information:
Unit 5- Geometry
Unit 6- Trigonometry
Angles
Proofs
1. Lines and Angles
a. Theorems include: vertical angles are congruent; when a
transversal crosses parallel lines, alternate interior angles are
congruent and corresponding angles are congruent; points on
a perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
2. Triangles
a. Theorems include: measures of interior angles of a triangle
sum to 180°; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is
parallel to the third side and half the length; the medians of a
triangle meet at a point, a line parallel to one side of a triangle
divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
b. Use Triangle Congruence Postulates including: SSS, SAS, ASA,
AAS, HL, CPCTC
3. Parallelograms
a. Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect
each other, and conversely, rectangles are parallelograms with
congruent diagonals.
1. Radian Measure vs. Degrees
a. Convert Degrees to Radians, multiply by
b. Convert Radians to Degrees, multiply by
1. Triangle AA similarity and sides are proportional
1. Know vocabulary and relationships between measured values.
a. Inscribed angle = measured arc length
b. Diameter = 2 x radius
2. Formulas:
#
2)
a. Arc Length =
$%&°
b.
Area of Sector =
#
$%&°
) *
Graphing Trig Functions
3. Graphing sin, cos, and tan functions. Use the graphing
calculator, change your mode to radians (most of the
time) and change your window using ZOOM, TRIG.
Identities
4. Know your reciprocal, tangent, cotangent, and
Pythagorean identities by heart. Sorry, no formula sheet
Other:
1. Arithmetic vs. Geometric
Volume and Surface Area Questions as Application
Circles
+&°
+&°
2. Use the Unit Circle or Calculator to find sin, cos, or tan of a
given angle. Make sure to convert your calculator’s
mode to Degrees from Radians.
Sequences and Series
Similarity
*
1. Know your formulas!!
Additional Information: