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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 a₀ 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 )