Algebra & Number Systems

TL;DR

You'll learn about different types of numbers and their properties, especially real numbers. You'll master polynomials, including finding their zeroes and applying key theorems like the Division Algorithm. Finally, you'll tackle pairs of linear equations, learning how to solve them graphically and algebraically.

1. The Mental Model

Think of numbers as building blocks and algebra as the rules for combining and manipulating these blocks. You're learning the fundamental language of math, which helps you describe relationships and solve problems. You'll move from basic number types to solving equations that model real situations.

2. The Core Material

Number Systems: The Real Numbers

You'll primarily work with Real Numbers. This huge group includes:
* Natural Numbers (N): Counting numbers (1, 2, 3, ...).
* Whole Numbers (W): Natural numbers plus zero (0, 1, 2, 3, ...).
* Integers (Z): Whole numbers and their negatives (... -2, -1, 0, 1, 2, ...).
* Rational Numbers (Q): Numbers that can be written as a fraction p/q, where p and q are integers and q is not zero (e.g., 1/2, -3, 0.75). Their decimal representations are either terminating or non-terminating repeating.
* Irrational Numbers: Numbers that cannot be written as p/q. Their decimal representations are non-terminating and non-repeating (e.g., $\sqrt{2}$, $\pi$).

Key Concept: The Fundamental Theorem of Arithmetic
Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. You use this for finding HCF and LCM.
* HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Found by taking the smallest power of each common prime factor.
* LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers. Found by taking the highest power of all prime factors involved.
* Relationship: For any two positive integers a and b, HCF(a, b) $\times$ LCM(a, b) = a $\times$ b.

Polynomials

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
* Degree: The highest power of the variable in a polynomial.
* Linear Polynomial: Degree 1 (e.g., ax + b)
* Quadratic Polynomial: Degree 2 (e.g., ax² + bx + c)
* Cubic Polynomial: Degree 3 (e.g., ax³ + bx² + cx + d)
* Zeroes of a Polynomial: The values of the variable for which the polynomial evaluates to zero. Graphically, these are the x-intercepts.
* A quadratic polynomial can have at most two zeroes.
* Relationship between Zeroes and Coefficients for a Quadratic Polynomial (ax² + bx + c):
* Sum of zeroes ($\alpha + \beta$) = -b/a
* Product of zeroes ($\alpha \beta$) = c/a

Division Algorithm for Polynomials: If P(x) and G(x) are two polynomials with G(x) $
e$ 0, then we can find polynomials Q(x) (quotient) and R(x) (remainder) such that:
P(x) = G(x) $\times$ Q(x) + R(x)
where R(x) = 0 or degree of R(x) < degree of G(x). This is just like division with numbers.

Pair of Linear Equations in Two Variables

A linear equation in two variables (like x and y) is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. The graph of such an equation is a straight line.

You'll deal with pairs of linear equations:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

Graphical Method of Solution:
* Plot both lines on the same graph.
* The point of intersection (x, y) is the solution.
* Types of Solutions:
* Intersecting lines: Unique solution (a₁/a₂ $
e$ b₁/b₂)
* Parallel lines: No solution (a₁/a₂ = b₁/b₂ $
e$ c₁/c₂)
* Coincident lines: Infinitely many solutions (a₁/a₂ = b₁/b₂ = c₁/c₂)

Algebraic Methods of Solution:
1. Substitution Method:
* Express one variable in terms of the other from one equation.
* Substitute this expression into the second equation.
* Solve for the remaining variable.
* Substitute back to find the first variable.

2. Elimination Method:

   • Multiply one or both equations by suitable non-zero constants so that the coefficients of one variable become numerically equal.
   • Add or subtract the equations to eliminate that variable.
   • Solve for the remaining variable.
   • Substitute back to find the eliminated variable.

3. Cross-Multiplication Method:

   • For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
     x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁)
   • Then, x = (b₁c₂ - b₂c₁) / (a₁b₂ - a₂b₁) and y = (c₁a₂ - c₂a₁) / (a₁b₂ - a₂b₁)
   • You must remember the formula. This method is often quicker once you're familiar with it.

3. Worked Example

Problem: Find the zeroes of the quadratic polynomial P(x) = 2x² - 8x + 6 and verify the relationship between the zeroes and the coefficients.

Solution:
1. Find the zeroes:
Set P(x) = 0:
2x² - 8x + 6 = 0
Divide by 2:
x² - 4x + 3 = 0
Factorize: Find two numbers that multiply to 3 and add to -4. These are -1 and -3.
(x - 1)(x - 3) = 0
So, x - 1 = 0 or x - 3 = 0
x = 1 or x = 3
The zeroes are $\alpha$ = 1 and $\beta$ = 3.

2. Verify relationship with coefficients:
   For P(x) = 2x² - 8x + 6, we have a = 2, b = -8, c = 6.

   • Sum of zeroes:
     $\alpha + \beta$ = 1 + 3 = 4
     Using formula: -b/a = -(-8)/2 = 8/2 = 4
     The sums match.

   • Product of zeroes:
     $\alpha \beta$ = 1 $\times$ 3 = 3
     Using formula: c/a = 6/2 = 3
     The products match.

The relationship is verified.

4. Key Takeaways

• Real numbers include natural, whole, integers, rational, and irrational numbers.
• The Fundamental Theorem of Arithmetic helps you understand prime factorization and calculate HCF/LCM.
• For a quadratic polynomial ax² + bx + c, the sum of zeroes is -b/a and the product is c/a.
• The Division Algorithm for polynomials is a structured way to divide them, yielding a quotient and remainder.
• A pair of linear equations can have a unique solution, no solution, or infinitely many solutions, depending on whether the lines intersect, are parallel, or are coincident.
• You can solve pairs of linear equations using graphical methods or algebraic methods (substitution, elimination, cross-multiplication).

Common Mistakes to Avoid:
- Forgetting to check the relationship between HCF, LCM, and the product of the two numbers.
- Mixing up the signs in the sum and product of zeroes formulas (especially -b/a).
- Making arithmetic errors when applying the algebraic methods for solving linear equations.
- Not understanding the conditions for unique, no, or infinite solutions for linear equation pairs.

5. Now Try It

Solve the following pair of linear equations using the elimination method:
3x + 4y = 10
2x - 2y = 2

What success looks like: You'll find the values of x and y that satisfy both equations, showing your steps clearly. The correct solution is x = 2 and y = 1.