Advanced Algebra

Rationalizing the Denominator

"Rationalizing the denominator" is when we move a root (like a square root or cube root) from the bottom of a fraction to the top. this is very similar to conjugate

Types of sequence

1. Arithmetic Sequences 2. Geometric Sequences 3. Triangular Numbers - is generated from a pattern of dots which form a triangle 4. Square Numbers, N^2 5. Fibonacci Sequence - The next number is found by adding the two numbers before it together (0, 1, 1, 2, 3....) some of these sequence are simply given as (check diagram)

Differentiation Arithmetic and Geometric Sequence

1. Check if the next term requires '+' or 'x' Arithmetic: +, -, addition, reduce Geometric: x, 'divide', increases by" Or "multiply" percentage: growth/ reduction, increase/ decrease Percentage Growth of 2.4% is 1.024

Behaviours of Series

1. Convergent - the sum goes to a specific number 2. Divergent

Synthetic Division - Process

1. Copy the coefficients from the terms in descending order. Use zeros for missing terms. 2. Find the root associated with the divisor. 3. Bring down the first coefficient. 4. Multiply the root value times the first coefficient and add it to the second coefficient. 5. Multiply the root value times this sum and add to the next coefficient. 6. Continue until the last coefficient is used. 7. Solution: *if there are 2 constants, divide the last figure by the divisor

Geometric Series

1. Look for words like "sum", "total", "how many" and "accumulation" over a duration, eg. For example: rain fall over 5 days VS rain fall on the 5th day

Multiplying and Dividing in the Polar Form

1. Multiply = add the angles 2. divide = subtract the angles

Geometric Sequence

1. This involves the "multiplication" of a constant 'r' to find the next term r - common ratio 2. this forms a curve that can be linked to power rules an some interest equations Arithmetic sequence forms a straight line

Solving Arithmetic Sequence

1. always find the first term. 2. at least 2 consecutive terms will be needed to find d (subtract one from the other ) 3. for 2 non consecutive terms this will lead to a simultaneous equation to find a and d 4. if n = 5, there are 6 terms

Arithmetic series

1. always try to find the 1st and the last term 2. in this series the last term is the 20th term Question: A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater?

Polynomials - Long Division (more than 1 variable)

1. choose a lead variable for the divisor 2. express the exponent of this variable in chronological order for the dividend

Polynomials - Long Division

1. ensure that the exponent of both terms are in chronological order 2. start with the first 'x' term of the divisor.

factoring polynomials

1. linear factors 2. Irreducible Quadratic

Identifying different series

1. look at the different layout of the series to see which one resembles Arithmetic or Geometric 2. Arithmetic usually has '+' or '-' 3. Geometric 'x'

multiplying complex numbers

1. remember that i squared is, -1 2. it is similar to binomial expansion

adding complex numbers

1. use surds to simplify the negative number

Calculating angles with Tan

1. use the horizontal line at the line of reference. 2. calculate the reference angles from this

Conjugate

1. when a binomial expression is multiplied by its conjugate the middle number is replaced. leaving only the square of the end terms. 2. this is linked to the differences of 2 squares 3. used to simplify the denominator to an integer (Real Number)

Geometric Sequence - trick 1

A ball is dropped from a height of 8 feet. The ball bounces to 80% of its previous height with each bounce. How high does the ball bounce on the fifth bounce? Round answer to the nearest tenth of a foot. Tip - the starting point is the first bounce.

Fundamental Theorem of Algebra

Any polynomial of degree n has n roots

Arithmetic Sequence

Each term after the first is found by "adding" a constant, called the common difference, to the previous term. "a" represents a term in the sequence for example: the last term or 7th term "d" common difference - the term that is added and it a constant, it doesn't change "n" - is the term number NB: know when to assume certain information for example: a year = 12 months, triangle anlges = 180 key words: increase or decrease in amount

Recursion Sequence

Each term of the sequence is defined as a function of the preceding term. Eg. Fibonacci Sequence, An = An-2 + An-1, recursive function.

finding the Degree

Find the degree for each term by adding the exponents of each variable in it

Arithmetic Sequence - trick 2

For 2 non consecutive terms - 1. Identify the smaller term as the An term 2. put this term in the equation of the other 3. to find n count the 1st term as 0. 4. equate this to the value of the last term given

Arithmetic Sequence - trick 1

For 2 non consecutive terms - 1. this will lead to a simultaneous equation to find a and d

Simplifying Polynomials

If it has different variables factor the power that will make it a quadratic equation.

Synthetic Division - Example 1

If there are missing values, replace them with 0

Triangle Sequence

N(N + 1)/ 2

Pascal Triangle

Pascal can be used once there are no coefficients in front of the binomials. for eg (a + b)

Irreducible Quadratic

That type of Quadratic (where we can't "reduce" it any further without using Complex Numbers)

Arithmetic Mean

The terms between any two non- successive terms of an arithmetic sequence. For example , if a sequence has non successive terms 30 and 74. The three terms in between are the arithmetic mean, that is 41, 52, 63 19, 30, 41, 52, 63, 74, 85, 96, ...

Geometric Sequence - trick 2

The third term of a geometric sequence is 3 and its sixth term is 1/9. Find the first term.

Synthetic Division - Purpose

These are used to evaluate a function where: 1. if the remainder is 0 (ie. y = 0), then the divisor is a root of the function 2. if the remainder is not 0, this is a point on the curve that does not lie on the x axis. f(3) = x^3 - 6x +4

Sum of an Infinite Series

These series have an infinite amount of terms, however if the rules below apply it will get closer to the sum figure. 1. the ratio (r) must be between -1 and 1 (like a fraction) for the sum to exist. 2. find the first term and use the formula if r > 1, it does not have a sum

Polynomial

They consist of constants, variables and exponents but: 1. never division by a variable. 2. does not have a fraction as an exponent

Synthetic Division

This is a "short-hand" version of long division for polynomials. Once one root is found by RRT, this can be used to simplify the polynomial which will help to find the other roots. the divisor will always look like this (x - a) where: 1. the coefficient and exponent of x is 1 They usually plot a line when graphed

Binomial Theorem

This is a polynomial with 2 terms. Eg. a and b, x and y. This is similar to the combination formula It is also linked to Pascals Triangle

Polar Form

This is another way to represent Complex numbers graphically. Question: Covert 3 + 4i to Polar Form 1. plot the point on the graph to get an idea where the angle is 2. Use Tan to find angle and ensure it is in the right quadrant 3. find the magnitude/ modulus

Complex numbers - powers

This is used when dealing with questions where the power is higher than 2

Purpose of Arithmetic Series

To find the "SUM" of a certain number of terms

The outcome of the roots

When the degree is odd (1, 3, 5, etc) there is at least one real root

Sequence vs Series

When the terms of a sequence are added together, the sum is referred to as a series.

Synthetic Division - Example 2

always ensure that the degrees are written in order

Binomial Theorem - tip 1

always raise the first term to (n - k) as it is the decreasing quantity. *this is the reverse position of binomial distribution NB - the nth term is always k - 1. therefore the 3rd term is k = 2 this is important when finding a specific term

check if a quadratic is irreducible

check if the discriminant is negative b^2 - 4ac

Polynomials - Long Division (missing term)

each term can be replaced with a 0

Converting Polar to Rectangular Form

equate both equations

Linear Factors

factors like (x-r1), because it makes a line when plotted

Arithmetic Sequence - trick 3

find 'd' with 3 consecutive terms that have variables. 1. subtract each term to find 'd' 2. equate the results and solve for the variable

Closure of Complex numbers

if the operation produces the same type of number. eg. a real number x a real number = real number

Square Number Sequence

n^2

radicals

surds

dividing Complex numbers B

the conjugate will turn the denominator into a real number

Binomial Theorem - tip 2

the entire polynomial must be raised to the exponential

Purpose of Arithmetic Sequence

the is used to find a term

Representation of Complex numbers

they can be represented in: 1. Vector/ rectangular Form 2. Polar Form 3. Exponential

Summation Notation

this is also called Sigma Notation

Rational Root Theorem (RRT)

this is done to narrow down the list of possible roots 1. write the quadratic in order of the exponents 2. look at factors of the constant (a0) and the leading coefficient (a n) 3. look at all possible fractional combination of the factors. a0/ a n factors of 3 is 3, 1. 4. This will give an answer, however you must put each in the equation to know which is the answer

dividing Complex numbers A

to simplify, multiply both denominator and numerator by 'i'

Complex numbers - squaring

when squared, the angle of a complex number is doubled. When multiplying two complex numbers in Polar Form, if a diagram is given: multiply the magnitudes, add the angles. This can be done with simple observation by looking at the vector lines

Complex numbers and Fundamental Theorem

you can use complex numbers to find the square root of a negative number