Algebraic expressions are mathematical phrases that involve numbers, variables, and operation symbols. They are used to represent relationships and perform calculations in algebra. An algebraic expression can include constants, coefficients, variables, and various mathematical operations such as addition, subtraction, multiplication, and division.
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An algebraic expression does not contain an equality sign; it represents a value rather than stating that two values are equal.
Expressions can be simplified by combining like terms or applying the distributive property.
An example of an algebraic expression is $$3x^2 + 5x - 7$$, which contains a polynomial with three terms.
Variables in algebraic expressions can take on different values, making these expressions versatile for modeling real-world situations.
Understanding how to manipulate algebraic expressions is crucial for solving equations and inequalities.
Review Questions
How do you simplify the algebraic expression $$4x + 3x - 2$$, and what steps do you take in this process?
To simplify the expression $$4x + 3x - 2$$, you combine like terms. First, you add the coefficients of the variable $$x$$: $$4x + 3x = 7x$$. The expression now reads $$7x - 2$$. This step demonstrates how combining like terms leads to a simpler form of the original algebraic expression.
What role do coefficients play in algebraic expressions, and how would you identify them in the expression $$5y^3 - 2y + 9$$?
Coefficients are the numerical factors that multiply the variables within an algebraic expression. In the expression $$5y^3 - 2y + 9$$, the coefficient of $$y^3$$ is $$5$$, while the coefficient of $$y$$ is $$-2$$. The number $$9$$ is considered a constant term since it does not multiply any variable.
Evaluate how manipulating algebraic expressions can aid in solving real-world problems and provide an example.
Manipulating algebraic expressions allows for the formulation of equations that model real-world scenarios. For instance, if a person wants to calculate their total cost for buying $$x$$ items at a price of $$p$$ each plus a fixed shipping fee of $$s$$, they would use the expression $$px + s$$. By simplifying or rearranging this expression, one can easily solve for different variables, helping to make informed decisions based on various purchasing scenarios.
Related terms
Variables: Symbols used to represent unknown values or quantities in mathematical expressions and equations.
Coefficients: Numerical factors that multiply the variables in an algebraic expression.
Polynomials: A specific type of algebraic expression that consists of multiple terms, where each term includes a variable raised to a non-negative integer exponent.