Glossary

5.1 Algebraic Expressions

3 min read•june 18, 2024

are the building blocks of math, using letters and symbols to represent numbers and operations. They're like a secret code that lets us describe relationships between quantities and solve complex problems.

Simplifying and performing operations are key skills for working with algebraic expressions. By combining , following the , and applying basic math rules, we can manipulate these expressions to find solutions and make calculations easier.

Algebraic Expressions

Translation of verbal to algebraic expressions

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  • Variables represent unknown or changing quantities in mathematical expressions
    • Common variables include xx, yy, aa, bb, etc. (zz, tt, nn)
  • Mathematical operations express relationships between quantities
    • Addition combines quantities using the ++ symbol (5+35 + 3)
    • Subtraction finds the difference between quantities using the −- symbol (8−28 - 2)
    • Multiplication scales quantities using the ×\times symbol or (2×4(2 \times 4 or 2(4))2(4))
    • Division splits quantities using the ÷\div symbol or a fraction bar (10÷210 \div 2 or 102\frac{10}{2})
  • Translating verbal descriptions to algebraic expressions involves identifying the unknown quantity, assigning a , and expressing the relationship using appropriate mathematical operations
    • "The difference between a number and 7" translates to x−7x - 7, where xx represents the unknown number
    • "Three times the sum of a number and 4" translates to 3(x+4)3(x + 4), where xx represents the unknown number
  • uses symbols and letters to represent numbers and operations in mathematical expressions

Simplification of algebraic expressions

  • Like have the same variables raised to the same powers and can be combined by adding or subtracting their coefficients
    • 2x22x^2 and −5x2-5x^2 are like terms, while 3y3y and 4y24y^2 are not
    • Combining like terms: 4a+3a=7a4a + 3a = 7a, −2b−5b=−7b-2b - 5b = -7b
  • The (PEMDAS) specifies the sequence in which to simplify expressions
    1. Parentheses: Simplify expressions inside parentheses first (2+(3×4))=(2+12)=14(2 + (3 \times 4)) = (2 + 12) = 14
    2. Exponents: Evaluate exponents, including powers and roots (23×4)=(8×4)=32(2^3 \times 4) = (8 \times 4) = 32
    3. Multiplication and Division: Perform from left to right (10÷2×3)=(5×3)=15(10 \div 2 \times 3) = (5 \times 3) = 15
    4. Addition and Subtraction: Perform from left to right (5+3−2)=(8−2)=6(5 + 3 - 2) = (8 - 2) = 6
  • Simplifying expressions using PEMDAS involves applying each step in the correct order
    • 3+2×(5−2)2−1=3+2×32−1=3+2×9−1=3+18−1=203 + 2 \times (5 - 2)^2 - 1 = 3 + 2 \times 3^2 - 1 = 3 + 2 \times 9 - 1 = 3 + 18 - 1 = 20

Operations with algebraic expressions

  • Addition and subtraction involve combining like terms and distributing negative signs
    • (5x−3)+(2x+4)=7x+1(5x - 3) + (2x + 4) = 7x + 1
    • (4a+2b)−(3a−b)=a+3b(4a + 2b) - (3a - b) = a + 3b
  • Multiplication uses the to multiply each of the first expression by each term of the second
    • (3x−2)(2x+1)=6x2+3x−4x−2=6x2−x−2(3x - 2)(2x + 1) = 6x^2 + 3x - 4x - 2 = 6x^2 - x - 2
    • (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2
  • Division involves dividing each term of the numerator by the denominator and simplifying the result
    • 10x2−5x5x=2x−1\frac{10x^2 - 5x}{5x} = 2x - 1
    • 6a2b+9ab23ab=2a+3b\frac{6a^2b + 9ab^2}{3ab} = 2a + 3b

Mathematical Symbols and Equations

  • are used to represent operations, relationships, and quantities in algebraic expressions
    • Equality symbol (=) indicates that two expressions have the same value
    • Inequality symbols (<, >, ≤, ≥) show the relationship between two expressions
  • An is a mathematical statement that asserts the equality of two expressions
    • Equations often involve variables and are used to solve for unknown values

Key Terms to Review (28)

Algebraic expressions: Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. They do not contain an equality sign, which differentiates them from equations.
Algebraic notation: Algebraic notation is a standardized way of writing mathematical expressions using symbols, numbers, and letters to represent variables and constants. It provides a clear and concise method for expressing mathematical ideas, enabling easy manipulation and communication of equations and formulas in algebra. This notation forms the backbone of algebraic expressions, allowing for operations such as addition, subtraction, multiplication, and division to be represented efficiently.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression or equation. It indicates how many times to scale the variable, influencing the overall value of the expression. Understanding coefficients is crucial in various mathematical contexts, as they help to establish relationships between variables and define equations in both algebraic and linear formats.
Constant: A constant is a fixed value that does not change, regardless of the variables around it. In mathematical expressions, constants provide stability and serve as reference points, allowing for clear definitions and calculations. They are essential in forming equations, as they help differentiate between what is variable and what remains unchanged.
Distributive Property: The distributive property is a fundamental algebraic principle that states that multiplying a number by a sum is the same as multiplying each addend separately and then adding the results. This property is essential for simplifying expressions and solving equations, and it connects to various mathematical concepts such as logic, operations with real numbers, manipulation of exponents, and algebraic expressions.
Equal sign: The equal sign (=) is a symbol used to indicate that two expressions have the same value. It is fundamental in forming equations and expressing mathematical equality.
Equation: An equation is a mathematical statement that asserts the equality of two expressions. It consists of two expressions separated by an equals sign (=).
Equation: An equation is a mathematical statement that asserts the equality of two expressions, typically involving variables and constants. Equations play a crucial role in solving problems and modeling real-world situations, allowing us to represent relationships between quantities. They can be simple or complex and often require manipulation to find unknown values.
Expansion: Expansion refers to the process of multiplying an algebraic expression by distributing each term across another expression, effectively rewriting it in an extended form. This technique helps clarify the relationships between the terms and can simplify the process of combining like terms or solving equations. By expanding expressions, one can reveal hidden structures and patterns, making it easier to manipulate and understand mathematical relationships.
Exponent: An exponent is a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. It plays a crucial role in simplifying complex calculations, allowing for the representation of large numbers and operations in a more compact form. Exponents are also essential in various mathematical concepts, including scientific notation, where they express values in terms of powers of ten, and in algebraic expressions, where they determine variable behaviors.
Expressions: An expression is a combination of numbers, variables, and operators (such as +, -, *, and /) that represents a mathematical relationship. Unlike equations, expressions do not include an equality sign (=).
Factoring: Factoring is the process of breaking down an algebraic expression into a product of simpler factors. This concept is essential for simplifying expressions, solving equations, and understanding the relationships between variables, especially in polynomial equations where the roots can be identified through factoring.
FOIL method: The FOIL method is a technique used for multiplying two binomials. The acronym FOIL stands for First, Outside, Inside, Last, which refers to the order in which you multiply the terms of the binomials. This method simplifies the multiplication process and helps in efficiently combining like terms to form a new polynomial expression.
Like Terms: Like terms are terms in an algebraic expression that have the same variable raised to the same power. They can be combined through addition or subtraction, making them crucial for simplifying expressions and solving equations. Identifying like terms is essential in algebra, as it helps in organizing expressions and performing calculations more efficiently.
Linear expression: A linear expression is a mathematical phrase that represents a straight line when graphed. It typically includes variables raised to the first power, constants, and coefficients, and can take the form of $$ax + b$$, where $$a$$ and $$b$$ are constants and $$x$$ is the variable. Linear expressions are foundational in algebra as they form the basis for linear equations and functions.
Mathematical Symbols: Mathematical symbols are standardized notations that represent numbers, operations, relationships, and functions in mathematics. These symbols are essential for expressing mathematical ideas clearly and concisely, enabling mathematicians and students to communicate complex concepts without ambiguity. Their use is crucial in algebraic expressions, where symbols convey operations like addition, subtraction, multiplication, division, and the relationships between variables.
Order of operations: Order of operations is a set of rules that specifies the correct sequence to evaluate a mathematical expression. It ensures consistency and avoids ambiguity in solving equations.
Order of Operations: The order of operations is a mathematical rule that dictates the sequence in which calculations should be performed to ensure consistent and correct results. This rule is essential in simplifying expressions, especially when they involve multiple operations like addition, subtraction, multiplication, division, and the use of parentheses. Understanding this concept is crucial for accurately solving problems involving rational numbers and algebraic expressions.
Parentheses: Parentheses are symbols used in mathematics to indicate that the operations contained within them should be performed before any outside operations. They play a critical role in establishing order and clarity in mathematical expressions, especially when multiple operations are involved. By grouping numbers or expressions with parentheses, one can alter the sequence of calculations, ensuring the intended interpretation of an expression is achieved.
Polynomial: A polynomial is an algebraic expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Common examples include linear equations and quadratic equations.
Polynomial: A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents. Polynomials can be represented in various forms, including as a sum of terms, where each term is a product of a coefficient and a variable raised to a power. They are foundational in algebra and play a significant role in understanding equations, particularly when solving for variables or analyzing relationships between different quantities.
Quadratic expression: A quadratic expression is a polynomial expression of degree two, typically written in the form $$ax^2 + bx + c$$, where 'a', 'b', and 'c' are constants, and 'x' is the variable. This type of expression represents a parabola when graphed and can have various applications in solving real-world problems, such as finding areas or modeling projectile motion.
Simplification: Simplification is the process of reducing expressions to their simplest form by eliminating unnecessary elements or combining like terms. This technique allows for easier manipulation and understanding of mathematical expressions, making it crucial in solving problems accurately and efficiently. Simplification is commonly applied to rational numbers and algebraic expressions, ensuring that calculations can be performed more easily.
Simplify an expression: Simplifying an expression means reducing it to its most basic form without changing its value. This involves combining like terms, factoring, and eliminating any unnecessary components.
Term: A term is a single mathematical expression that can be a number, a variable, or the product of numbers and variables. In algebraic expressions, terms are the building blocks that come together to form more complex structures, allowing for mathematical operations and manipulations. Each term is separated by a plus or minus sign, highlighting its individual contribution to the overall expression.
Terms: Terms are individual components separated by addition or subtraction in an algebraic expression. Each term can include numbers, variables, or both, possibly multiplied together.
Variable: A variable is a symbol, typically a letter, that represents an unknown quantity in mathematical expressions and equations. Variables allow us to create general rules and relationships in mathematics, enabling the representation of multiple values without specifying them directly. They play a critical role in forming algebraic expressions and solving linear equations, as they can change or vary based on the context or specific problem at hand.
Zero Product Property: The Zero Product Property states that if the product of two or more factors equals zero, then at least one of the factors must be equal to zero. This principle is crucial for solving equations, particularly quadratic equations, as it allows for the determination of variable values when the equation is set to zero.
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