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Algebraic Foundations

• Fully worked topic solutions are available online.

Overview

• In 2017, the Cassini spacecraft disintegrated in Saturn's atmosphere after a successful 7-year mission to observe the planet and its moons involving 22 orbits.
• Earlier, the Voyager 1 spacecraft passed Pluto in 1990 and left the solar system in 2004 with a time capsule of information about Earth.
• Mathematics and physics are essential for space missions.
• Mathematics is proposed as the universal language for potential communication with extraterrestrial intelligence, citing the universality of Pythagoras' theorem (a² + b² = c²) across all galaxies.
• Expressing Pythagorean relation in formulas exemplifies the succinct nature of algebra.
• Elementary algebra establishes fundamental skills, while advanced algebras like group theory are used in internet security.
• Cheryl Praeger, from Australia, and the late Maryam Mirzakhani are modern mathematicians who are known for their work.

Algebraic Skills

• By the end of this subtopic, students will be able to expand, factorize, and simplify expressions, including algebraic fractions.
• The topic emphasizes algebraic skills required for learning and understanding more complex math. The distributive law is fundamental for expansion.

Review of expansion and factorisation

• Distributive law: a(b+c) = ab + ac.
• Simple expansions include: (a+b)(c+d) = ac + ad + bc + bd, (a+b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b², and (a + b)(a – b) = a² – b².

Expanding and Simplifying Expressions

• Expanding $2(4x - 3)^2 – (x - 2)(x + 2) + (x + 5)(2x – 1)$ involves expanding brackets, expanding fully while taking care with signs, collecting like terms, and stating the answer (coefficient of x is -39).

Factorisation

• Some simple factors include common factors, the difference of two perfect squares, perfect squares and other quadratic trinomials.
• A systematic approach to factorising is via identifying common factors and number of terms.
• Terms that can be grouped are commonly referred to as grouping ’2 and 2’ or grouping ’3 and 1’.
• If the first three terms of a² + 2ab + b² – c² are a perfect square grouping ‘3 and 1’ creates a difference of two squares allowing expression to be factorised.

Factorising Algebraic Expressions

• 3x² - 12y² can be factorised by first taking out the common factor, recognising the difference of two squares, and then factorising to get 3(x - 2y)(x + 2y).
• x² + 7x-8 needs the numbers that multiply to the product and add to middle. The middle term must be split. The expression is factorised as (x-1)(x + 8).
• To factorise 3x² - 17x + 10 find two numbers multiply to the outside and add to the middle and factorise to get = (3x-2) (x - 5).
• 2x3 + 5x2y - 12xy² take out the common factor and then find the numbers to multiply find the product, express as x (2x – 3y) (x + 4y).

Factorising Sums and Differences of Perfect Cubes

• (a + b)(a² – ab + b²) = a³ + b³, sum of two cubes.
• (a - b)(a² + ab + b²) = a³ – b³, difference of two cubes.

Factorising Perfect Cubes

• x³ – 27 can be expressed as x³ – 3³ and factorised using the appropriate rule.
• 2x³ + 16 equals can be simplified as = 2(x + 2)(x² – 2x + 4).

Algebraic Fractions

• Simplifying, adding, subtracting, multiplying and dividing algebraic fractions is done using the same methods as with arithmetic fractions.
• An algebraic fraction can be simplified by cancelling any common factor between numerator and denominator.

Simplifying Algebraic Fractions

• $\frac{x^2-2x}{x^2-5x+6}$ can be simplified by factorising both numerator and denominator and cancelling the common factor leading to $\frac{x}{x-3}$.
• $\frac{x^4-1}{x-3} \div \frac{1+x^2}{3-x}$ can be simplified by changing the division into multiplication through replacing divisor as a reciprocal, factorising where possible to help create common factors of both numerator and the denominator and getting simplifying terms = $1-x^2$.

Adding and Subtracting Algebraic Fractions

• When adding or subtracting algebraic fractions, factorisation and expansion techniques are often required.
• Denominators should be factored and each fraction expressed with the lowest common denominator; the terms are to be simplified in the numerator.

Pascal's Triangle and Binomial Expansions

• By the end of this subtopic, students should be able to expand, simplify perfect cubes and use binomial equations from Pascals triangle.

Expansions of Perfect Cubes

• The perfect square (a + b)² may be expanded quickly by the rule (a + b)² = a² + 2ab + btical Methods. The perfect cube (a + b)³ can also be expanded by a rule.
• (a + b)³ = a³+3a²b+3ab² + b³.
• (a – b)³ = a³ – 3a²b + 3ab² – b³.

Features of the rule for expanding perfect cubes

• The powers of the first term, a, decrease as the powers of the second term, b, increase.
• The coefficients of each term in the expansion of (a + b)³ are 1, 3, 3, 1.
• The coefficients of each term in the expansion of (a – b)³ are 1, −3, 3, −1.
• The signs alternate + in the expansion of (a – b)³.

Expanding a Perfect Cube

• (2x-5)³ equals 8x³ – 60x² + 150x – 125.

Pascal’s Triangle

• Pascal's triangle contains patterns; each row begins and ends with ‘1’, and other numbers are formed by adding two terms.

Using Pascal's Triangle in Binomial Expansions

• For (a - b)^5 the power of binomial is 5. The binomial coefficents are 1, 5, 10, 10, 5, 1. Alternated to give equation down to the terms.
• To expand (2x – 1)^5 can be expresses as 32x^5 − 80x⁴ + 80x³ – 40x² + 10x – 1.

The Binomial Theorem

• By the end of the lesson you should be able to evaluate expressions with factorial equations, calculate combinations, expand using the binomial theorem and determine particular terms with the theorem.

Factorial Notation

• Calculations like 7 × 6 × 5 × 4 × 3 × 2 × 1 can be shortened using a factorial key on many calculators. Definition: is defined by n! = n × (n-1) × (n − 2) × ... ×3×2×1.

Evaluating an Expression with Factorial Notation

• $5! - 3! + \frac{50!}{49!}$ is evaluated by expanding the two smaller factorials, writing the larger factorial and simplifying then calculating the final answer = 164.

Formula for binomial coefficients

• Terms can be expressed in factorial notation. $C = \frac{n!}{r! (n-r)!}$
• are called combinatoric coefficients.

Pascal's triangle with combinatoric coefficients

• Pascal's triangle can be expressed using combination rules, which can be seen through multiple row and column arrangements.
• Binomial expansions can be expressed using this notation for each of binomial coefficients.

Patterns

• (ⁿ₀) = 1 = (ⁿₙ) (the start and end of each row of Pascal's triangle)
• (ⁿ₁) = n = (ⁿₙ₋₁) (the second from the start and the second from the end of each row)
• (ⁿᵣ) = (ₙ₋ᵣ)

Evaluating Combinations

• Combination is evaluated through identifying the formula to use, writing large with terms of largest and simplifying them and calculating answer through simplification.
• For ${8 \choose 3}$ has value of = 56

The Binomial Theorem

• The binomial equation can be formed using the binomial pattern, through determining (x + y)", and setting out as required in order with combinations.

The General Term of the Binomial Theorem

• For the expansion for ( x + y )^n -there are terms ( n + 1 )
• General rules -( ⁿ ₀ ) ⁿ^n, ( ⁿ₁ ) x^(n–1)y …. (ⁿᵣ)x^(n–r)
• Expansion of a (x - y) is similar to (x + y) although the signs change for each term.

Determining a Term in the Binomial Expansion

• The general formula can determine which term has special functions. ${ t }_{ r+1 } = \begin{pmatrix} n \ r \end{pmatrix} { \left( \frac{x}{2} \right) }^{ n-r }{ \left( { -\frac{y}{3} } \right) }^{ r }$

Identifying a Term in the Binomial Expansion

• Identify what term would countain x^8 while expanding, write the generic expansion and reaggrage it express in the coefficient.

Sets of Real Numbers

-Classify number as element of the real system, as well as to use and write in interval notion.

• The concept of numbers is counting and the introduction of symbols.
• Technologies have been devised to assist in counting and computational techniques.
• Real are the number that are +, zero or negative.

Set Notation

• Sets are a collection of objects which are elements of a set. Denoted through symbols in the set. .
• The universal are { 1,2,3,4,5,6,7,8,9,10 }, can include subsets of A,B and C. The table below gives the common symbols used in set notation. Symbol: ∈, the set of as, ∉: not an element , ⊂: subclass of, U: reunion and intersection.
• The empty/desjoint sets have not thing in common. ∅ set .

Classifications/Rules

• Although coutning member are to solve equation then need to be a fraction.
• The set of numbers are shown in the table through our real number system. The following numbers are also shown: N:{1,2,3,4}, Z ,Q,E and R.
• The relationships between in which are seen through the diagram is the relations and types of integers.

The numbers that are real numbers

• R has positive, 0 or negative. Where R∞∪ {0} ∪R +

The numbers that don't have real number

• It in improtant. To recognize the number that are ∞ -Any form a/v does any form to reperenst a division.

Interval notation

• Interval notation provides an alternative way. -{a,b} is where a is great or = and x.

Worked examples .

• Express a number on the number line and alterntive notation.
• Us the interval notations.

Surds

• Students should be able to simplify and order them as well as expand and simplify equations with them, at the rationalise dominatiors with needs.
• Surds are the ^nth root ^n√x where x is any irrational; decimal value is approximate.
• Surds have radicals signs but are surds; they cvanoti valued; x√26 is but √25 cannot since 5.

Ordering Surds

• Surds are real number with positions on number; it is place with positions if number and place with perpect squares.
• For example √6 it is between rational by 6 the perfect squares 4<6<9 √4<√6 < 09 There for be √6< 3
• The symbols used in the worked examples are √≈2, 44.

Simplest forrm

• All have no perpect square factors and have simples form under, radical; in which simples form when the square root sign not cant find. For example, 3√5 is a form a √45 but 5√3 is the form for 75.

Example

√128 = √64 × (2). = √ x + √x (2) = 8√2 x ∴√ = 8√√2

• It obey number that we have in multiplication sign. The adductions signs should follow .√c, (x+by) can be equal if and only if √c

Expansions

Expansions of brackets containing surds are worked out with distributive law. √(a) =u, and all like terms

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