Review of Fundamental Algebraic Concepts

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in the form a × 10n, where 1 ≤ |a| < 10 and n is an integer.

• Purpose: Simplifies calculations and representation of numbers in science and engineering.

• Conversion to Scientific Notation: Move the decimal point so that only one nonzero digit remains to the left of the decimal. Count the number of places moved to determine the exponent.

• Example: 45,000 = ; 0.00032 =

Addition, Subtraction, and Multiplication of Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

• Addition/Subtraction: Combine like terms (terms with the same variable and exponent).

• Multiplication: Use the distributive property or special products (e.g., ).

• Example:

• Example:

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors.

• Common Methods: Factoring out the greatest common factor (GCF), factoring trinomials, difference of squares, and grouping.

• Example: (difference of squares)

• Example: (factoring out GCF)

Multiplication and Division of Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials.

• Multiplication: Multiply numerators together and denominators together, then simplify.

• Division: Multiply by the reciprocal of the divisor.

• Example:

• Example:

Addition and Subtraction of Rational Expressions

To add or subtract rational expressions, first find a common denominator.

• Steps:

  1. Factor denominators if possible.

  2. Find the least common denominator (LCD).

  3. Rewrite each expression with the LCD.

  4. Add or subtract numerators, keep the denominator.

  5. Simplify if possible.

• Example:

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree (the highest exponent of the variable is 1).

• General Form:

• Solution:

• Example:

Solving Quadratic Equations by Factoring

Quadratic equations are equations of the form .

• Factoring: Write the quadratic as a product of two binomials and set each factor to zero.

• Example:

Using the Quadratic Formula

The quadratic formula provides solutions to any quadratic equation.

• Formula:

• Example: or

Addition, Subtraction, and Multiplication of Complex Numbers

Complex numbers are numbers of the form a + bi, where i is the imaginary unit ().

• Add/Subtract: Combine real parts and imaginary parts separately.

• Multiply: Use distributive property and .

• Example:

• Example:

Solving an Equation by Factoring

Factoring can be used to solve equations by setting each factor equal to zero.

• Example:

Solving Inequalities

Inequalities involve expressions related by <, ≤, >, or ≥.

• Linear Inequality: Solve as you would an equation, but reverse the inequality sign when multiplying or dividing by a negative number.

• Example:

Graphs and Equations of Lines

Finding the Equation of a Line (Point-Slope Form)

The point-slope form is useful for finding the equation of a line when you know a point and the slope.

• Formula: , where is a point on the line and is the slope.

• Example: Find the equation of a line through (2, 3) with slope 4:

Finding the Equation of Parallel and Perpendicular Lines

Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals.

• Parallel: If a line has slope , any parallel line has slope $m$.

• Perpendicular: If a line has slope , any perpendicular line has slope (provided ).

• Example: Given , a parallel line through (0, 3):

• Example: A perpendicular line through (1, -2): Slope is , so