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Math 41 Study Guide

4.1 Polynomial Functions and their Graphs

• End behavior:

o Look at leading coefficient/exponent and check sign

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

Graphing a polynomial :

o Factor

o Find x- and y-intercepts

o Find end behavior

o 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
where p is a factor of the constant coefficient and q is a factor of the leading

• How to find all zeros of a polynomial:

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

o List all possible rational zeros using the Rational Zeros Theorem

o Test the possible zeros

o If you find a zero, factor it out

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

o n > m: no horizontal asymptote

o n = m: horizontal asymptote is

o n < m: horizontal asymptote is y = 0

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

• Graphing rational functions:

o Factor numerator and denominator

o Find x- and y-intercepts

o Find horizontal and vertical asymptotes

o 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) = ax, memorize the graph:

o Horizontal asymptote y = 0

o no vertical asymptote

o Domain = (−∞,∞)

o Range = (0,∞)
Compound interest formula:

• Continuously compounded interest: A(t) = ert

5.2 Logarithmic Functions

• Definition of logarithm:

• Properties:

, memorize the graph:

o Vertical asymptote: x = 0

o no horizontal asymptote

o Domain = (0,∞)

o Range = (−∞,∞)

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

Common log :

• Natural log:

5.3 Laws of Logarithms

• no laws for

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

5.4 Exponential and Logarithmic Equations

Solving exponential equations:

o Isolate the exponential term on one side

o 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

o Pull the exponent to the front and solve the equation

• Solving logarithmic equations:

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

o Isolate the logarithmic term on one side

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

o Use the property to get rid of the log

o Solve the equation

Two special cases of exponential equations:

o Combination of exponential and polynomial terms: try to factor

o Sum of multiple exponential terms: try to use substitution

5.5 Modeling with Exponential and Logarithmic Functions

• Exponential growth model:

• To solve any problem you usually have to find and r

• Formulas and logarithmic scales

6.1 Angle Measure

• Relationship between Degrees and Radians:

o convert from degrees to radians by multiplying by 
o 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: v = rω

6.2 Trigonometry of Right Triangles

• Trigonometric Ratios:

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

• Solving right triangles

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