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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
- 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.
- Subtract and bring down the next term.
- 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
- Use the inverse of “c” as the divisor, and write only the coefficients of the polynomial.
Don’t forget placeholders for missing variable.
- First number comes straight down.
- Multiply that number and the divisor together, and place under the next number.
- Add the two numbers together. Repeat until there are no numbers left.
- If you end up with a number other than zero, this will be your remainder.
- 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:
- Build an xy table.
- Plug in at least 3 values for x (You might
need more than 3 values)
- Find the y - values.
- 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
- 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
