Algebra: Key Concepts

Algebra 1

Perfect Squares

Expansion

Dividing Polynomials

(x±a)^2

Difference of Two Squares

Factorising Quadratics

-b formula

Binomial Expansion

binomial theorem

Algebraic Identities and Factors

Solving Linear Equations

Solving Simultaneous Linear Equations

2 Variables "x+y"

3 Variables "x+y+z"

Algebra 2

Quadratic Equations

How to solve

-b formula

graph

factorise

substitution

Nature of Quadratic Roots in ax^2+bx+c

The Discriminant

(b^2-4ac)

Real and Distinct

Complex/ Imaginary

Rational

Real and Equal

(b^2-4ac) > 0

(b^2-4ac) < 0

= to a perfect square

(b^2-4ac) = 0

Solving Quadratic + Linear Equations

Substitution

If only 1 line intersects, it's a tangent

Forming Quadratics from their Roots

x^2 - x(sum of roots) + (product of roots) = 0

Completing the Square from a Quadratic x^2+bx+c

(x + b/2)^2 - (b/2)^2 + c

Max. & Min.

Written as a(x - p)^2 + q = min. point (p,q)

Written as q - a(x - p)^2 = max. point (p,q)

Algebraic Surd Equations

1 Surd

2 Surds

Isolate
Square
Solve
Substitute

Move one to each side
Square both sides
Isolate any remaining surds
Square again
Solve
Substitute

The Factor Theorem

If f(k) = 0, then (x - k) is a factor,
Conversely, if (x - k) is a factor, then f(k) = 0
If (ax - k) is a factor, then f(k/a) = 0

Factorising Cubics

Solving Cubics

Trial and error in f(x) until it = 0 and divide

Find factors, equate to 0 and then find roots

Algebra 3

Inequalities

Symbols

less than/equal to ≤

more than/equal to ≥

less than <

more than >

Quadratic Inequalities

ax^2 + bx + c ≥ 0

solve to find real roots, draw a rough sketch, use graph to find set of values that satisfy the inequality

Graph

If a > 0, graph is U-shaped

If a < 0, graph is ⋂-shaped

Rational Inequalities

(( 3x - 2 ) / (x + 1 )) ≥ 0

Multiply both sides by (x + 1)^2 to retain inequality as any number squared is always positive ----> multiply out both sides and solve quadratic inequality

Modulus

If ∣x∣ = a, then x = ±a and ∣x∣^2 = a^2

Modular Inequalities

If ∣x∣ < 1 then x lies between 1 and -1.... i.e -1 < x < 1

If ∣x∣ > 1 then x lies outside.... i.e x > 1 or x < -1

Middle Test

To check whether x is inside or outside, get the middle number between the two x values of a quadratic and sub in

PROOFS

Direct Proof

Proof by Contradiction

Axioms, definitions and other proven theorems are used to verify statements

We show that if a statement is to be true, then a logical contradiction occurs and so it proves the original statement to be false

Prove that √2 is irrational

Assume it's rational, if rational then it can be written as a/b (a,b ∈ Z) and b=/= 0, and a and b have no common factors)

square both sides, a^2 is even since it's equal to 2b^2, a is even, b is even, let a = 2c

CONTRADICTION: a and b have common factor 2 so √2 is irrational

Proofs of Abstract Inequalities

If a and b are real numbers, then a^2 ≥ 0 and b^2 ≥ 0

(a + b)^2 ≥ 0

(a - b)^2 ≥ 0

-(a + b)^2 ≤ 0

-(a - b)^2 ≤ 0

Indices

4^2

4 = base number

2 = power/index

Logarithmic Function

The log of a number is the power to which the base number must be raised to get that number

a^x = y is the same as loga y = x

Types of logs

loge = natural

log10 = common

the log of a number to its own base = 1

Solving log functions

check that each term has the same base, if not, the change of base rule must be used to change to a common base,,, check solutions, as negative logs aren't defined

ln x^n = n.lnx

ln e = 1

INDUCTION: NOT COVERED AS OF 24/5/22