Lecture Note in Mathematics

Comprehensive Algebra Review Notes

Welcome to this comprehensive guide on Algebra. This lecture note covers essential algebraic concepts and methods, from basic algebraic expressions to solving quadratic equations. It's designed to help you understand fundamental principles, work through examples, and prepare for algebra assessments. Whether you're revisiting foundational topics or preparing for an exam, these notes will provide clear explanations, step-by-step solutions, and practical examples.

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1. Basic Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. Unlike an equation, it does not include an equals sign and cannot be solved, but it can be simplified or evaluated.

1.1 Components of an Algebraic Expression

• Constants: Fixed numbers (e.g., \(3\), \(-7\), \( \frac{1}{2} \)).
• Variables: Symbols that represent unknown values, often denoted by letters such as \( x \), \( y \), or \( z \).
• Coefficients: Numbers that multiply a variable (e.g., in \(5x\), \(5\) is the coefficient).
• Operators: Symbols that indicate operations, such as \( + \), \( - \), \( \times \), and \( \div \).
• Terms: Parts of an expression separated by \( + \) or \( - \) signs. Each term can consist of a constant, a variable, or a product of both (e.g., \(3x\), \(-5y\), \(7\)).

1.2 Types of Algebraic Expressions

• Monomial: An expression with only one term, such as \( 4x \) or \( -5y^2 \).
• Binomial: An expression with two terms, such as \( x + 5 \) or \( 3y - 2y^2 \).
• Trinomial: An expression with three terms, such as \( x^2 + 4x + 4 \).
• Polynomial: An expression with multiple terms, such as \( x^3 - 2x^2 + 5x - 3 \).

1.3 Examples of Algebraic Expressions

Example 1

Simplify the expression:

\( 3x + 2x \)

Solution: Combine like terms by adding the coefficients of \( x \):

\( 3x + 2x = 5x \)

Example 2

Identify the terms, coefficients, and constant in the expression:

\( 4x^2 - 3x + 7 \)

• Terms: \( 4x^2 \), \( -3x \), \( 7 \)
• Coefficients: \( 4 \) (for \( x^2 \)), \( -3 \) (for \( x \))
• Constant: \( 7 \)

Example 3

Evaluate the expression when \( x = 2 \):

\( 5x + 3 \)

Solution: Substitute \( x = 2 \) and solve:

\( 5(2) + 3 = 10 + 3 = 13 \)

1.4 Simplifying Algebraic Expressions

To simplify an algebraic expression, combine like terms (terms with the same variables and exponents) and perform operations.

Example 4

Simplify the expression:

\( 2x + 3x^2 + 4x - 5 \)

Solution: Combine the like terms for \( x \):

\( 3x^2 + (2x + 4x) - 5 = 3x^2 + 6x - 5 \)

1.5 Key Points to Remember

• Expressions with only numbers (e.g., \(7\) or \(3 - 5\)) are called numerical expressions.
• Expressions with variables (e.g., \(4x\), \(x + 3\)) are called algebraic expressions.
• Only like terms can be combined (terms with the same variable and exponent).
• Expressions can be simplified but not solved unless set equal to something (e.g., an equation).

This overview of basic algebraic expressions covers the essential concepts and provides practice for identifying terms, simplifying, and evaluating expressions.

2. Solving Linear Equations

A linear equation is an equation where the highest power of the variable is one. Linear equations often represent a straight line when graphed, and they generally have one solution.

2.1 General Form of a Linear Equation

The general form of a linear equation in one variable \( x \) is:

\( ax + b = 0 \)

where:

• \( a \) and \( b \) are constants, with \( a \neq 0 \) (if \( a = 0 \), the equation would not contain a variable).

2.2 Steps to Solve a Linear Equation

To solve for \( x \), isolate the variable on one side of the equation. This can be done by performing the following steps:

• Step 1: Move the constant term (\( b \)) to the other side of the equation by adding or subtracting.
• Step 2: Divide by the coefficient of \( x \) (\( a \)) to solve for \( x \).

This process results in:

\( x = -\frac{b}{a} \)

2.3 Example Problems

Example 1

Solve the equation:

\( 3x + 6 = 0 \)

Solution:

• Step 1: Subtract 6 from both sides to isolate the term with \( x \):
\( 3x = -6 \)
• Step 2: Divide by 3 to solve for \( x \):
\( x = -2 \)

Example 2

Solve the equation:

\( -4x + 8 = 0 \)

Solution:

• Step 1: Subtract 8 from both sides:
\( -4x = -8 \)
• Step 2: Divide by -4 to solve for \( x \):
\( x = 2 \)

Example 3

Solve the equation involving fractions:

\( \frac{2x}{3} - 5 = 1 \)

Solution:

• Step 1: Add 5 to both sides:
\( \frac{2x}{3} = 6 \)
• Step 2: Multiply both sides by 3 to clear the fraction:
\( 2x = 18 \)
• Step 3: Divide by 2 to solve for \( x \):
\( x = 9 \)

2.4 Special Cases

Case 1: No Solution

If the equation simplifies to a false statement, such as \( 0 = 5 \), it has no solution (inconsistent equation).

Example:

\( 2x + 3 = 2x - 4 \)

Subtract \( 2x \) from both sides:

\( 3 = -4 \) (false statement)

Case 2: Infinite Solutions

If the equation simplifies to a true statement, such as \( 0 = 0 \), it has infinite solutions (identity equation).

Example:

\( 4x + 5 = 4x + 5 \)

Subtract \( 4x \) from both sides:

\( 5 = 5 \) (true statement)

2.5 Key Points to Remember

• A linear equation has at most one solution unless it’s an identity (infinite solutions) or inconsistent (no solution).
• Always isolate the variable by reversing operations around it.
• Check for special cases by simplifying the equation whenever possible.

3. Quadratic Equations

A quadratic equation is a second-degree equation, meaning it has a variable raised to the power of two (e.g., \( x^2 \)). It is generally represented in the standard form:

\( ax^2 + bx + c = 0 \)

where:

• \( a \), \( b \), and \( c \) are constants, with \( a \neq 0 \) (if \( a = 0 \), the equation would be linear).

3.1 Solutions to Quadratic Equations

Quadratic equations can have two solutions, one solution, or no real solution, depending on the discriminant (the part under the square root in the quadratic formula).

The quadratic formula, used to solve any quadratic equation, is:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

3.2 Discriminant and Types of Solutions

The discriminant \( \Delta = b^2 - 4ac \) helps determine the type of solutions:

• If \( \Delta > 0 \): Two distinct real solutions.
• If \( \Delta = 0 \): One real solution (repeated root).
• If \( \Delta < 0 \): No real solutions (two complex solutions).

3.3 Example Problems

Example 1

Solve the quadratic equation:

\( x^2 - 5x + 6 = 0 \)

Solution:

• Identify \( a = 1 \), \( b = -5 \), and \( c = 6 \).
• Calculate the discriminant:
\( \Delta = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1 \)
• Since \( \Delta > 0 \), there are two distinct real solutions.
• Apply the quadratic formula:
\( x = \frac{-(-5) \pm \sqrt{1}}{2 \cdot 1} = \frac{5 \pm 1}{2} \)
• Simplify to get the solutions:
\( x = 3 \) and \( x = 2 \)

Example 2

Solve the quadratic equation:

\( 2x^2 - 4x + 2 = 0 \)

Solution:

• Identify \( a = 2 \), \( b = -4 \), and \( c = 2 \).
• Calculate the discriminant:
\( \Delta = (-4)^2 - 4 \cdot 2 \cdot 2 = 16 - 16 = 0 \)
• Since \( \Delta = 0 \), there is one real solution (repeated root).
• Apply the quadratic formula:
\( x = \frac{-(-4) \pm \sqrt{0}}{2 \cdot 2} = \frac{4 \pm 0}{4} = 1 \)
• The solution is \( x = 1 \).

Example 3

Solve the quadratic equation:

\( x^2 + x + 1 = 0 \)

Solution:

• Identify \( a = 1 \), \( b = 1 \), and \( c = 1 \).
• Calculate the discriminant:
\( \Delta = (1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \)
• Since \( \Delta < 0 \), there are no real solutions. The solutions are complex.
• Apply the quadratic formula:
\( x = \frac{-1 \pm \sqrt{-3}}{2 \cdot 1} = \frac{-1 \pm i\sqrt{3}}{2} \)
• The complex solutions are \( x = \frac{-1 + i\sqrt{3}}{2} \) and \( x = \frac{-1 - i\sqrt{3}}{2} \).

3.4 Key Points to Remember

• Use the quadratic formula to solve any quadratic equation.
• The discriminant helps determine the type of solutions: real and distinct, real and repeated, or complex.
• In some cases, quadratic equations can also be solved by factoring or completing the square.

4. Polynomials

A polynomial is an expression consisting of variables, coefficients, and exponents combined with addition, subtraction, and multiplication.

For example:

\( f(x) = 4x^3 - 3x^2 + 2x - 5 \)

The degree of a polynomial is the highest power of the variable.

5. Factoring

Factoring is the process of breaking down a polynomial into simpler terms (factors) that, when multiplied together, give the original polynomial.

Common methods of factoring include:

• Factoring out the Greatest Common Factor (GCF)
• Factoring trinomials
• Difference of squares: \( a^2 - b^2 = (a + b)(a - b) \)

6. Inequalities

An inequality compares two values, expressions, or quantities, showing that one is larger or smaller than the other.

Symbols used:

• \( < \): Less than
• \( > \): Greater than
• \( \leq \): Less than or equal to
• \( \geq \): Greater than or equal to

Example: Solve \( 2x + 3 > 7 \)

\( x > 2 \)

7. Rational Expressions

A rational expression is a fraction where the numerator and/or the denominator are polynomials.

For example:

\( \frac{3x + 2}{x^2 - 4} \)

To simplify, factor and reduce where possible:

\( \frac{3x + 2}{(x+2)(x-2)} \)

8. Exponents and Powers

Exponents represent repeated multiplication of a base. For example:

\( a^n = a \cdot a \cdot \ldots \cdot a \) (n times)

Rules of exponents:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \), where \( a \neq 0 \)

9. Radicals

A radical expression involves a root, such as a square root or cube root.

Notation:

\( \sqrt{a} \) is the square root of \( a \), and \( \sqrt[n]{a} \) is the nth root of \( a \).

Rules of radicals:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)
• \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \), where \( b \neq 0 \)

10. Absolute Value

The absolute value of a number represents its distance from zero on a number line, always yielding a non-negative value.

Notation: \( |x| \)

Example:

\( | -5 | = 5 \) and \( | 5 | = 5 \)

Properties:

• \( |a \cdot b| = |a| \cdot |b| \)
• \( |a + b| \leq |a| + |b| \) (Triangle Inequality)