# Algebra Study Guide

**4.1 Polynomial Functions and their Graphs
**• End behavior:

◦ Look at leading coefficient/exponent and check sign

◦ If polynomial is factored, check sign of each factor and multiply

• Graphing a polynomial:

◦ Factor

◦ Find x- and y-intercepts

◦ Find end behavior

◦ Either use test points between the intercepts or memorize the shape around
zeros

depending on the multiplicity:

— If multiplicity is 1, then it crosses the x-axis in a
straight line

— If multiplicity is even, then it turns back around

— If multiplicity is odd > 1, then it ”squiggles” through the x-axis

**4.2 Dividing Polynomials
**• Long Division: Make sure to fill in missing powers

• Synthetic Division: Only works for division by (x − c).
Again make sure to fill in 0’s

for missing powers

• Remainder Theorem: to find P(c) carry out a synthetic
division for c, the remainder

is P(c)

• Factor Theorem: c is a zero of P ↔ (x − c) is a factor of P(x)

**4.3 Real Zeros of Polynomials
**• Rational Zeros Theorem: The possible rational zeros of a polynomial are of
the form

p/q where p is a factor of the constant coefficient a

_{0}and q is a factor of the leading

coefficient a

_{n}

• How to find all zeros of a polynomial:

◦ Try previous factoring methods like substitution or
grouping, if this does not work

then:

◦ List all possible rational zeros using the Rational Zeros Theorem

◦ Test the possible zeros

◦ If you find a zero, factor it out

◦ Repeat from the top until your polynomial is quadratic, then factor/complete
the

square/quadratic formula

**4.4 Complex Zeros and the Fundamental Theorem of
Algebra
**• Fundamental Theorem of Algebra: every polynomial of degree n has precisely
n zeros

(zeros of multiplicity k are counted k times)

• Conjugate Zeros Theorem: If a complex number is a zero
of polynomial with real

coefficient, then its conjugate is also a zero

**4.5 Rational Functions
**• Horizontal asymptotes: n is the degree of the numerator, m is the degree
of the denominator

◦ n > m: no horizontal asymptote

◦ n = m: horizontal asymptote is

◦
n < m: horizontal asymptote is y = 0

• Vertical asymptotes: zeros of the denominator (that do not cancel with the numerator)

• Graphing rational functions:

◦ Factor numerator and denominator

◦
Find x- and y-intercepts

◦
Find horizontal and vertical asymptotes

◦
Either use test points between intercepts/vertical asymptotes or use the shape

around vertical asymptotes/intercepts to determine the shape of the graph

• Slant asymptote: only exists if the degree of the
numerator is one greater than the

degree of the denominator: use long/synthetic division

**5.1 Exponential Functions
**• f(x) = a

^{x}, memorize the graph:

◦ Horizontal asymptote y = 0

◦
no vertical asymptote

◦
Domain = (−∞,∞)

◦
Range = (0,∞)

• Compound interest formula:

• Continuously compounded interest:

**5.2 Logarithmic Functions
**• Definition of logarithm: log

_{b}a = x ↔ b

^{x}= a

• Properties:

◦ log_{b} 1 = 0

◦
log_{b} b = 0

◦
log_{b} b^{x} = x

◦

• f(x) = log_{b} x, memorize the graph:

◦ Vertical asymptote: x = 0

◦
no horizontal asymptote

◦
Domain = (0,∞)

◦
Range = (−∞,∞)

• Finding the domain of logarithmic function: logarithms only defined for positive numbers

• Common log:

• Natural log:

**5.3 Laws of Logarithms
**• log

_{b}(xy) = log

_{b}x + log

_{b}y

•

•

• no laws for log_{b}(x + y) or log_{b} x · log_{b} y

• Change of base: where c can be any positive number

**5.4 Exponential and Logarithmic Equations
**• Solving exponential equations:

◦
Isolate the exponential term on one side

◦
Take logarithm of both sides:

— If there is only one exponential term, use that base for the log

— If there is an exponential term on both sides, use either the common or
natural

log

◦ Pull the exponent to the front and solve the equation

• Solving logarithmic equations:

◦
If there are multiple logarithmic terms, combine them into one using logarithmic

laws

◦
Isolate the logarithmic term on one side

◦
Raise the base of the logarithm to the left and the right side of the equation

◦
Use the property
to get rid of the log

◦
Solve the equation

• Two special cases of exponential equations:

◦
Combination of exponential and polynomial terms: try to factor

◦
Sum of multiple exponential terms: try to use substitution

**6.1 Angle Measure
**• Relationship between Degrees and Radians:

◦
convert from degrees to radians by multiplying by

◦ convert from radians to degrees by multiplying by

• Coterminal angles: Angle between 0° and 360° degrees (or 0 and 2π )

• Length of a circular arc: s = rθ (θ in rad)

• Area of a circular sector: (θ in rad)

• Linear Speed and Angular Speed: and

• Relationship between linear and angular speed:

**6.2 Trigonometry of Right Triangles
**• Trigonometric Ratios:

• Values of the trig ratios for angles 30° , 45° and 60°

• Solving right triangles

**6.3 Trigonometric Functions of Angles
**• Memorize in which quadrants each trig function is positive

• Reference angles: Acute angle formed by x-axis and terminal side

• Using reference angles to evaluate trig functions

• Reciprocal Identities:

• Pythagorean Identities:

• Expressing trig functions in terms of other trig functions

• Evaluating trig functions using identities

• Area of a Triangle: 1/2ab sinθ (where θ is the angle between a and b)

**6.4 Law of Sines
**• Law of Sines:

• Solving triangles:

◦
SAA

◦
SSA (either no solution, one solution or two solutions)

**6.5 Law of Cosines
**• Law of Cosines:

◦ a

^{2}= b

^{2}+ c

^{2}− 2bc cosA

◦ b

^{2}= a

^{2}+ c

^{2}− 2ac cosB

◦ c

^{2}= a

^{2}+ b

^{2}− 2ab cosC

Solving triangles:

◦
SSS

◦
SAS

• Navigation: Bearing

• Heron’s Formula: Area of a triangle is