Algebraic Expressions

Simplifying algebraic expressions
Like terms, collecting or combining like terms
Evaluating algebraic expressions

Terms, variables, constants - coefficients
An algebraic expression is one or more algebraic terms containing variables and constants connected by mathematical operations.
Terms are the elements separated by the plus or minus signs.
In algebraic expressions, variables are letters, such as a,  bc, or x, yz, that can have different values.
Constants are the terms or elements represented only by numbers.
Coefficients are the number part of the terms that multiply a variable or powers of a variable.
 Example: a2 - 6a + 5   - is the algebraic expression of three terms, where, a is the variable, - 6 is the coefficient of a of the middle term, and 5 is the constant.
Monomial, binomial, trinomial,  . . . polynomial
An algebraic expression consisting of a single term is called a monomial, expression consisting of two terms is binomial, three terms trinomial and an expression with more than three terms is called polynomial.
A monomial is a term containing variables with only nonnegative integers as exponents.
 Examples: -3x2  and  2ab  are monomials,     a3 - b3  is the binomial, x2 - 4x + 4  is the trinomial, and   a3 - 3a2 + 3a - 1  is the polynomial.
Simplifying algebraic expressions
By simplifying an algebraic expression, we mean reducing it in the simplest possible form which mainly involves: multiplication and division, removing (expanding) brackets and collecting (adding and subtracting) like terms.
Like terms are those terms which contain the same powers of same variables and which can only differ in coefficients.
 Examples: a)   - 4a3 + 3a2 + 5a3 - 7a2 = (- 4 + 5) · a3 + (3 - 7) · a2  = a3 - 4a2, b)   (x2  - x + 1) · (x + 1) = x3 - x2 + x + x2  - x + 1 = x3 + 1.
Evaluating algebraic expressions
To evaluate an algebraic expression means to replace (substitute) the variables in the expression with numeric values that are assigned to them and perform the operations in the expression.
 Example: Evaluate the expression  x2  - 6xy + 9y2  for x = 2  and  y =  -1. Solution: 22  - 6 · 2 · (- 1)  + 9 · (-1)2 = 4 + 12 + 9 = 25.
Beginning Algebra Contents B