Algebraic Expressions and Simplification

Basic Operations with Numbers and Variables

Algebraic expressions involve numbers, variables, and arithmetic operations. Simplifying these expressions is a foundational skill in precalculus, requiring the application of arithmetic rules, properties of exponents, and order of operations.

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

• Multiplication: Use the distributive property and properties of exponents.

• Division: Simplify fractions and reduce to lowest terms.

• Order of Operations: Follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

• Example: Simplify First, calculate the exponent: Then, inside p

• arentheses: Multiply: Add:

Fractions and Least Common Multiple (LCM)

Operations with Fractions

When adding or subtracting fractions, it is necessary to find a common denominator, often the least common multiple (LCM) of the denominators.

• LCM: The smallest positive integer that is a multiple of two or more numbers.

• Example: The LCM of 18 and 12 is 36.

• Adding Fractions: Find LCM (36), convert fractions:

Opposites and Reciprocals

Definitions and Examples

• Opposite: The opposite of a number is .

• Reciprocal: The reciprocal of a nonzero number is .

• Example: The opposite of $19-19\frac{5}{2}\frac{2}{5}$.

Solving Linear Equations

Isolating the Variable

To solve linear equations, isolate the variable on one side using inverse operations.

• Example: Subtract from both sides: Divide by :

Exponents and Radicals

Properties and Simplification

Exponents and radicals are used to express repeated multiplication and roots. Simplifying expressions with exponents often involves applying the laws of exponents.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Negative Exponent:

• Radical to Exponent:

• Example:

Polynomials: Addition, Subtraction, and Multiplication

Combining and Expanding Polynomials

Polynomials are algebraic expressions with terms consisting of variables raised to whole-number exponents. Operations include addition, subtraction, and multiplication.

• Standard Form: Write terms in descending order of exponents.

• Example:

• Multiplying Binomials:

Factoring Polynomials

Common Methods

Factoring is the process of expressing a polynomial as a product of its factors. Common methods include factoring out the greatest common factor, difference of squares, and the AC method for trinomials.

• Difference of Squares:

• Trinomial Factoring: where and are numbers such that and

• Example:

Solving Quadratic Equations

Factoring and the Quadratic Formula

Quadratic equations are equations of the form . They can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring: Set each factor equal to zero and solve for .

• Quadratic Formula: For , the solutions are:

• Example: Solve using the quadratic formula: , ,

Summary Table: Key Algebraic Properties