



Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Algebra formula sheet with polynomial, synthetics divisions, radicals, roots, rational exponents, complex numbers and distance formulas.
Typology: Cheat Sheet
1 / 6
This page cannot be seen from the preview
Don't miss anything!
Polynomials:
𝑛
𝑛
𝑛− 1
𝑛− 1
2
2
1
0
Standard form: a polynomial written with descending powers of the variable.
Monomial: polynomial with one term 5 𝑥
Binomial: polynomial with two unlike terms 10 𝑥
2
Trinomial: polynomial with three unlike terms 10 𝑥
2
Degree of the polynomial is the highest powered variable.
Ex. – 𝑥
7
5
2
Degree = 7.
Polynomial Long Division:
3
2
2
2 3 2
3
2
2
first term in the dividend. Then multiply
the result with the second term in the
divisor.
Synthetic Division:
Can be used as a shortcut when a polynomial is divided by 𝑥 − 𝑐
3
2
2
48
𝑥− 3
Don’t forget placeholders for missing variable.
Rational Exponents:
1
𝑛
= √𝑎
𝑛
𝑚
( √
𝑛
𝑚
Product and Quotient Rules for Radicals:
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
Radicals and Roots:
𝑛
If “a” is a positive real number and “n” is
even, then “a” has exactly two real n
th
roots Ex. √ 81 = ± 9
If “a” is any real number and “n” is odd,
then “a” has only one real root Ex. √ 27
3
If “a” is a negative real number and “n” is
even, then “a” has no real root. Ex. √− 64
Rationalizing the Denominator:
3
3
3
3
3
3
3
Adding and Subtracting Radicals:
When adding or subtracting radicals, the
number under the radical and root must be
the same, just like when combining variables.
Ex. √ 3 + √ 3 = 2 √ 3
(Remainder is written over the divisor.)
Multiplying and Dividing Radicals:
When multiplying and dividing radicals, do so
just like you would variables.
Ex. 3 √ 2 × 5 √ 6 = 15 √ 12
Complex Numbers:
In order to take square roots of negative
number, the number 𝑖 is used.
𝑖 = √− 1 Standard form:
2
3
= −𝑖 The conjugate of 𝑎 + 𝑏𝑖 is 𝑎 − 𝑏𝑖.
4
= 1 This is important for solving
equations with complex
numbers.
Distance Formula:
Finding the distance between two points
1
1
) and
2
2
2
1
2
2
1
2
Ex. ( 1 , 2 ) ( 5 , 9 )
2
2
2
2
Midpoint Formula:
Finding the midpoint of (𝑥 1
1
) and
2
2
1
2
1
2
Ex. ( 1 , 5 ) ( 4 , 7 )
Sketching a Graph:
need more than 3 values)
Draw the line through those points.
Discriminant of a Quadratic Equation:
If 𝑏
2
− 4 𝑎𝑐 > 0 , two unequal real
solutions.
If 𝑏
2
− 4 𝑎𝑐 = 0 , a repeated real solution
with a root of multiplicity 2.
If 𝑏
2
− 4 𝑎𝑐 < 0 , no rea solution.
Useful Equations:
Interest
Distance
Compound Interest
𝑛𝑡
Continuous Compounding Interest
𝑟𝑡
Inequality Intervals:
a
b
a
b
a
b
a
b
a
a
b
b
Solving Absolute Value Equations:
Absolute value equations can be split into two
equations.
Ex.
Absolute Value:
This |𝑎| denotes absolute value, which is the
distance a number is from the origin, 0, on the
number line. Ex.
Solving Inequalities:
Treat {<, >, ≤, ≥} like {=}. Except flip the sign
when × 𝑜𝑟 ÷ by a negative number.
Ex. 4 − 𝑥 ≥ 5 → −𝑥 ≥ 1 → 𝑥 ≤ − 1
Symmetry:
With respect to x-axis
With respect to y-axis
With respect to the origin
Graph of the square function:
2
Graph of the cube function:
3
Graph of the square root function:
Graph of the cube root function:
3
Graph of the reciprocal function:
Graph of the absolute value function:
Zeros of a function:
A polynomial can have as many zeros as its
degree.
Rational zeros theorem:
List all 𝑝, which are the factors of the constant
term, and all 𝑞, which are the factors of the
constant of the highest degreed variable.
𝑝
𝑞
are all of the possible zeros of the function.
Rational functions:
Form: 𝑅(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
Asymptotes:
The zeros of the denominator are the vertical
asymptotes of the function.
Horizontal asymptotes:
If the degree of the numerator < denominator,
then 𝑦 = 0 is a horizontal asymptote.
If the degree of the numerator ≥
denominator, first use long division to obtain
𝑔(𝑥)
𝑞(𝑥)
, then the zeros of 𝑞(𝑥)
are the asymptotes.
If 𝑓
= 𝑏, then 𝑦 = 𝑏 is a horizontal
asymptote.
If 𝑓
= 𝑎𝑥 + 𝑏, then 𝑦 = 𝑎𝑥 + 𝑏 is an
oblique asymptote.
All other cases, no asymptotes.
Composite function:
Plug 𝑔(𝑥) everywhere there is an x in 𝑓(𝑥)
Ex. 𝑓(𝑥) = 5 𝑥
2
2
Complex zeros:
If 𝑎 + 𝑏𝑖 is a complex zero, then the conjugate
𝑎 − 𝑏𝑖 is also a zero.
One-to-one function:
If every element in the domain corresponds to
a unique element in the range, then the
function is said to be one-to-one.
Horizontal line test:
Determines a one-to-one function like the
vertical line test.
Inverse Function:
First must be one-to-one.
Definition: 𝑓(𝑓
− 1
− 1
Given 𝑓
= 𝑎𝑥 + 𝑏, replace 𝑓
with 𝑦
and interchange 𝑥 and 𝑦. Solve for 𝑦.
Ex. 𝑓(𝑥) = 3 𝑥 + 5 𝑦 = 3 𝑥 + 5
Exponential Function:
𝑥
𝑒 is irrational and often the base of the
exponential function, but not always.
If 𝑎
𝑣
𝑢
, then 𝑣 = 𝑢
Uninhibited growth/decay:
0
𝑘𝑡
0
Logarithmic functions:
𝑦 = log 𝑎
𝑥 if and only if 𝑥 = 𝑎
𝑦
𝑦 = 𝑙𝑛𝑥 if and only if 𝑥 = 𝑒
𝑦
𝑦 = log 𝑥 if and only if 𝑥 = 10 ^𝑦
Systems of Equations:
Two equations with two unknowns. Can be
solved two ways 1. Elmination
Properties of logarithms:
log 𝑎
= log
𝑎
𝑀 + log
𝑎
log
𝑎
= log
𝑎
𝑀 − log
𝑎
log
𝑎
𝑟
= 𝑟 log
𝑎
𝑥
𝑥 ln 𝑒
ln 𝑒 = 1
ln
log 𝑎
log
𝑎