Algebraic Expressions
Algebraic Expressions
Evaluating Expressions
The foundation of algebra is the variable, an unknown number represented by a symbol (usually a letter such as x or a). Variables can be preceded by a coefficient, which is a constant in front of the variable, such as 4x or −2a.
An algebraic expression is any sum, difference, product, or quotient of variables and numbers (such as 2x + 7y − 1). The value of an expression is found by replacing the variable with a given value and simplifying the result.
Find the value of x 2 + 3 when x = 5.
x 2 + 3
(5)2 + 3 = 28
Adding and Subtracting Expressions
2x + 3xy – 2z + 6y + 2xy → 2x – 2z + 6y + (3xy + 2xy) → 2x – 2z + 6y + 5xy
Distributing and Factoring
Simplifying expressions may require distributing and factoring, which are opposite processes.
To distribute, multiply the term outside the parentheses by each term inside the parentheses. For each term, coefficients are multiplied, and exponents are added (following the rules of exponents).
2x (3x2 + 7) = 6x3 + 14x
Factoring is the reverse process: taking a polynomial and writing it as a product of two or more factors. The first step in factoring a polynomial is always to “undistribute,” or factor out, the greatest common factor (GCF) among the terms. The remaining terms are placed in parentheses
14a2 + 7a = 7a (2a + 1)
You can check your factoring by redistributing the term outside the parentheses.
To multiply binomials (expressions with two terms), use FOIL: First – Outer – Inner – Last. Multiple the first term in each expression, the outer terms, the inner terms, and the last term in each expression. Then simplify the expression.
(2x + 3)(x – 4)
= (2x)(x) + (2x)( –4) + (3)(x) + 3(–4)
= 2x2 – 8x + 3x –12
= 2x2 – 5x – 12
When variables occur in both the numerator and the denominator of a fraction, they cancel each other out. So, a fraction with variables in its simplest form will not have the same variable on the top and bottom.