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Mathematics college algebra formula sheet, Cheat Sheet of Algebra

Algebra formula sheet with polynomial, synthetics divisions, radicals, roots, rational exponents, complex numbers and distance formulas.

Typology: Cheat Sheet

2021/2022

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MATH 133 College Algebra
Polynomials:
𝑎𝑛𝑥𝑛+𝑎𝑛−1𝑥𝑛−1+𝑎2𝑥2+𝑎1𝑥+𝑎0
Standard form: a polynomial written with descending powers of the variable.
Monomial: polynomial with one term 5𝑥
Binomial: polynomial with two unlike terms 10𝑥2+5𝑥
Trinomial: polynomial with three unlike terms 10𝑥2+5𝑥+4
Degree of the polynomial is the highest powered variable.
Ex. 𝑥7+16𝑥5+5𝑥24𝑥+12
Degree = 7.
Polynomial Long Division:
3𝑥3+4𝑥2+𝑥+7÷𝑥2+1
3𝑥+4
7431 232 xxxx
−(3𝑥3 + 3𝑥)
4𝑥2 2𝑥 + 7
−(4𝑥2 + 4)
−2𝑥3
1. Divide the first term in the divisor by the
first term in the dividend. Then multiply
the result with the second term in the
divisor.
2. Subtract and bring down the next term.
3. Repeat until you can’t divide any more.
Synthetic Division:
Can be used as a shortcut when a polynomial is divided by 𝑥𝑐
2𝑥3𝑥2+3÷𝑥3
3 2 1 + 0 + 3
+6 + 15+45
2+5+15+48
2𝑥2+5𝑥+15+48
𝑥−3
1. Use the inverse of “c” as the divisor, and write only the coefficients of the polynomial.
Don’t forget placeholders for missing variable.
2. First number comes straight down.
3. Multiply that number and the divisor together, and place under the next number.
4. Add the two numbers together. Repeat until there are no numbers left.
5. If you end up with a number other than zero, this will be your remainder.
6. When rewriting, begin with 1 degree less than when you started.
Rational Exponents:
𝑎1
𝑛=𝑎
𝑛
𝑎𝑚
𝑛=(𝑎
𝑛)𝑚
Product and Quotient Rules for Radicals:
𝑢𝑣
𝑛=𝑢
𝑛𝑣
𝑛
𝑢
𝑣
𝑛=𝑢
𝑛
𝑣
𝑛
Radicals and Roots: 𝑎
𝑛
If “a” is a positive real number and “n” is
even, then “a” has exactly two real nth
roots Ex. 81=±9
If “a” is any real number and “n” is odd,
then “a” has only one real root Ex. 27
3=3
If “a” is a negative real number and “n” is
even, then “a” has no real root. Ex. 64
Rationalizing the Denominator:
3
5=3
5×5
5=15
5
4
9
3=4
9
3×3
3
3
3=43
3
27
3=43
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=23
(Remainder is written over the divisor.)
pf3
pf4
pf5

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MATH 133 College Algebra

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

+ 𝑥 + 7 ÷ 𝑥

2

2 3 2

x  x  x  x 

3

2

2

  1. Divide the first term in the divisor by the

first term in the dividend. Then multiply

the result with the second term in the

divisor.

  1. Subtract and bring down the next term.
  2. Repeat until you can’t divide any more.

Synthetic Division:

Can be used as a shortcut when a polynomial is divided by 𝑥 − 𝑐

3

2

+ 3 ÷ 𝑥 − 3

2

48

𝑥− 3

  1. Use the inverse of “c” as the divisor, and write only the coefficients of the polynomial.

Don’t forget placeholders for missing variable.

  1. First number comes straight down.
  2. Multiply that number and the divisor together, and place under the next number.
  3. Add the two numbers together. Repeat until there are no numbers left.
  4. If you end up with a number other than zero, this will be your remainder.
  5. When rewriting, begin with 1 degree less than when you started.

Rational Exponents:

1

𝑛

= √𝑎

𝑛

𝑚

𝑛

( √

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:

  1. Build an xy table.
  2. Plug in at least 3 values for x (You might

need more than 3 values)

  1. Find the y - values.
  2. Plot the points

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 < 𝑥 ≤ 𝑏

a

( ]

b

[

a

[ )

b

a

b

[𝑎, ∞) 𝑥 ≥ 𝑎

a

[

a

(−∞, 𝑏] 𝑥 ≤ 𝑏 ]

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

  1. Substitution

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