Algebraic Expressions

Algebraic Expressions

Definition

• An algebraic expression is a mathematical phrase that combines numbers, variables, and mathematical operations.
• It can be a single term or a combination of terms separated by operators (+, -, x, /).

Types of Algebraic Expressions

• Monomials: Expressions with a single term, e.g., 2x, 5y, 3z
• Binomials: Expressions with two terms, e.g., 2x + 3, x^2 - 4
• Polynomials: Expressions with three or more terms, e.g., x^2 + 3x - 2, 2x^3 - 5x^2 + x - 1

Operations with Algebraic Expressions

• Addition and Subtraction: Combine like terms by adding or subtracting their coefficients.
• Multiplication: Multiply each term in one expression by each term in the other expression.
• Distributive Property: Expand products by multiplying each term in the parentheses by the factor outside, e.g., 2(x + 3) = 2x + 6

Simplifying Algebraic Expressions

• Combine like terms: Combine terms with the same variable(s) and coefficient(s).
• Remove parentheses: Use the distributive property to expand products.
• Combine constants: Combine numeric terms.

Evaluating Algebraic Expressions

• Replace variables with given values to evaluate the expression.
• Follow the order of operations (PEMDAS): parentheses, exponents, multiplication and division, addition and subtraction.

Algebraic Expressions

• An algebraic expression is a mathematical phrase combining numbers, variables, and mathematical operations.
• It can be a single term or a combination of terms separated by operators (+, -, x, /).

Types of Algebraic Expressions

• Monomials are expressions with a single term, such as 2x, 5y, or 3z.
• Binomials are expressions with two terms, such as 2x + 3 or x^2 - 4.
• Polynomials are expressions with three or more terms, such as x^2 + 3x - 2 or 2x^3 - 5x^2 + x - 1.

Operations with Algebraic Expressions

• To add or subtract algebraic expressions, combine like terms by adding or subtracting their coefficients.
• To multiply algebraic expressions, multiply each term in one expression by each term in the other expression.
• The distributive property allows expanding products by multiplying each term in the parentheses by the factor outside, e.g., 2(x + 3) = 2x + 6.

Simplifying Algebraic Expressions

• Combine like terms to simplify algebraic expressions.
• Remove parentheses by using the distributive property to expand products.
• Combine constants by adding or subtracting numeric terms.

Evaluating Algebraic Expressions

• Replace variables with given values to evaluate the expression.
• Follow the order of operations (PEMDAS): parentheses, exponents, multiplication and division, addition and subtraction.

Quadratic Equations

• A quadratic equation is a type of algebraic expression that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• The highest power of the variable (x) in a quadratic equation is 2.

Key Characteristics

• The graph of a quadratic equation is a parabola that opens upwards or downwards.
• Quadratic equations can be factored into the product of two binomials.

Types of Quadratic Equations

• Monic quadratic equations have a = 1, such as x^2 + 5x + 6 = 0.
• Non-monic quadratic equations have a ≠ 1, such as 2x^2 + 3x + 1 = 0.

Methods for Solving Quadratic Equations

• Factoring involves expressing the equation as a product of two binomials and solving for x.
• The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
• Graphing involves finding the x-intercepts of the parabola by graphing the equation.

Quadratic Formula

• The discriminant, b^2 - 4ac, determines the number of solutions.
• If b^2 - 4ac > 0, there are two distinct real solutions.
• If b^2 - 4ac = 0, there is one real solution (repeated root).
• If b^2 - 4ac < 0, there are no real solutions (complex solutions).

Applications of Quadratic Equations

• Quadratic equations can be used to model projectile motion.
• They can be used to solve optimization problems, finding the maximum or minimum value of a quadratic function.
• Quadratic equations are applied in physics and engineering to describe the motion of objects and systems.