Materials algebra

ALGEBRA MATERIALS

Mathematics Handbook for Junior and Senior High School

Academic Year: 2024

TABLE OF CONTENTS

1. Introduction to Algebra

2. Algebraic Forms and Operations

3. Linear Equations and Inequalities

4. System of Linear Equations

5. Quadratic Equations and Functions

6. Exponents and Logarithms

7. Algebraic Fractions

8. Factoring Polynomials

CHAPTER I: INTRODUCTION TO ALGEBRA

1.1 Definition of Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. The fundamental concept of algebra is the use of variables, constants, and coefficients to express mathematical relationships.

1.2 Basic Components

In algebraic expressions, we recognize several important components:

– Variable: A symbol, usually a letter such as x, y, a, b, which represents an unknown value or can change value.

– Constant: A value that is fixed and does not change.

– Coefficient: A number that multiplies a variable.

– Term: A part of an expression separated by addition (+) or subtraction (-) signs.

Example:

In the expression 3x^2 – 5y + 7:

– Variables: x, y

– Coefficients: 3, -5

– Constant: 7

– Terms: 3x^2, -5y, 7

CHAPTER II: ALGEBRAIC FORMS AND OPERATIONS

2.1 Similar and Dissimilar Terms

– Similar Terms: Terms that have the same variables and the same powers/exponents.- Example: 2x and 5x; 3a^2 and -a^2.

– Dissimilar Terms: Terms that have different variables or different powers.- Example: 2x and 3y; 4a and a^2.

2.2 Addition and Subtraction

Addition and subtraction can only be performed on similar terms.

Example:

5a + 3a – 2b = (5 + 3)a – 2b = 8a – 2b

2.3 Multiplication and Division

Multiplication involves distributing the multiplication process to each term (distributive property), while division follows the rules of fraction simplification.

Properties:

– Commutative: a times b = b times a

– Associative: (a times b) times c = a times (b times c)

– Distributive: a times (b + c) = ab + ac

CHAPTER III: LINEAR EQUATIONS AND INEQUALITIES

3.1 Linear Equations in One Variable (LEOV)

A linear equation is an equation where the highest power of the variable is 1.

General Form:

ax + b = 0

Where a neq 0.

Steps for Completion:

1. Move variables to one side and constants to the other side.

2. Simplify the equation.

3. Find the value of the variable.

Example:

Solve 2x – 4 = 10

2x = 10 + 4

2x = 14

x = 7

3.2 Linear Inequalities

Inequalities use signs such as <, >, leq, geq. A crucial rule is that multiplying or dividing both sides by a negative number reverses the inequality sign.

CHAPTER IV: SYSTEM OF LINEAR EQUATIONS

4.1 System of Linear Equations in Two Variables (SLETV)

This system consists of two or more linear equations that work together.

General Form:

$$ begin{cases} a_1x + b_1y = c_1 a_2x + b_2y = c_2 end{cases} $$

4.2 Methods of Completion

There are three standard methods to solve this system:

1. Graph Method: Plotting both lines on a Cartesian plane. The intersection point is the solution.

2. Substitution Method: Replacing one variable with an expression from the other equation.

3. Elimination Method: Removing one variable by equalizing the coefficients.

CHAPTER V: QUADRATIC EQUATIONS AND FUNCTIONS

5.1 Definition

A quadratic equation is an equation where the highest exponent of the variable is 2.

General Form:

ax^2 + bx + c = 0

Where a, b, c are real numbers and a neq 0.

5.2 Roots of Quadratic Equations

The values of x that satisfy the equation are called roots. There are three types of roots based on the Discriminant (D):

D = b^2 – 4ac

– If D > 0: Two distinct real roots.

– If D = 0: One identical real root (twin root).

– If D < 0: No real roots (imaginary roots).

5.3 Formulas for Roots

Roots can be found using:

1. Factoring

2. Completing the perfect square

3. ABC Formula:

x_{1,2} = frac{-b pm sqrt{b^2 – 4ac}}{2a}

CHAPTER VI: EXPONENTS AND LOGARITHMS

6.1 Exponents

Exponentiation is a mathematical operation involving multiplication of the same number repeatedly.

Properties:

– a^m times a^n = a^{m+n}

– a^m : a^n = a^{m-n}

– (a^m)^n = a^{m times n}

– a^0 = 1

– a^{-n} = frac{1}{a^n}

6.2 Logarithms

Logarithm is the inverse operation of exponentiation.

Definition:

^a log b = c iff a^c = b

Properties:

– ^a log (b times c) = ^a log b + ^a log c

– ^a log (frac{b}{c}) = ^a log b – ^a log c

– ^a log b^n = n times ^a log b

CHAPTER VII: ALGEBRAIC FRACTIONS

7.1 Simplification

Algebraic fractions are simplified by factoring the numerator and denominator, then eliminating identical factors.

Example:

$$ frac{x^2 – 9}{x + 3} = frac{(x + 3)(x – 3)}{x + 3} = x – 3 $$

7.2 Operations

– Addition/Subtraction: Must equate the denominators first using the Least Common Multiple (LCM).

– Multiplication: Multiply numerator with numerator, denominator with denominator.

– Division: Change division to multiplication by reversing the second fraction.

CHAPTER VIII: FACTORING POLYNOMIALS

8.1 Common Factor

ab + ac = a(b + c)

8.2 Grouping Method

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

8.3 Quadratic Form ax^2 + bx + c

To factor x^2 + bx + c, find two numbers whose sum is b and product is c.

Special Formulas:

– a^2 + 2ab + b^2 = (a + b)^2

– a^2 – 2ab + b^2 = (a – b)^2

– a^2 – b^2 = (a + b)(a – b)

Prepared in 2024

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