Algebraic Expressions

Term: a number, variable, or a product or quotient of numbers and variables

Examples: 5, x, -3xy,

Numbered part of the term is called the numerical coefficient or coefficient. For the term –3xy, 3 is the coefficient

Like Terms: variable parts of the terms are exactly the same; i.e. same variable, same

exponent

Examples:

(a) 4ab and –3ab

(b) and

Unlike Terms: variable parts of the terms differ; i.e. different variables or the same variable

With different exponents

Examples:

(a) 5a and 7b

(b) and –10ab

Evaluating Algebraic Expressions

Order of Operations

(1) Perform operations inside the symbols of grouping-parentheses, brackets, fraction

bars, etc.

(2) Evaluate powers of numbers

(3) Do all multiplications and divisions in order from left to right

(4) Do all additions and subtractions in order from left to right

Examples:

(a)

(b)

To evaluate any algebraic expression replace the variable by its numerical value and follow the order of operations.

Examples: Evaluate each expression for the given values of the variables

(a)

(b)

Polynomials

Monomial: an algebraic expression containing one term: ex.

Binomial: an algebraic expression containing two terms: ex. 3x – 5

Trinomial: an algebraic expression containing three terms: ex.

Polynomial: an algebraic expression containing more than three terms: ex.

Addition and Subtraction of Algebraic Expressions

To add or subtract algebraic expressions: we add or subtract like terms by combining the numerical coefficients and attaching the variable part of the term to the answer.

Examples:

(a)

(b)

(c)

(d) (Remember the definition of Subtraction)

(e)

Problems

Combine

(1)

(2)

(3)

(4)

(5)

Answers

(1)

(2)

(3)

(4)

(5)

Multiplication of Algebraic Expressions

(1) Multiplying Two or More Monomials

Follow the laws of signed numbers and the laws of exponents

Examples:

(a)

(b)

(2) Multiplying a Monomial and a Polynomial

Multiply each term of the polynomial by the monomial

Examples:

(a)

(b)

(c)

(3) Multiplying a Polynomial and a Polynomial

Multiply each term of one polynomial by each term of the other polynomial and

combine the like terms.

Examples:

(a)

(b)

(c)

Observe:

(d)

(e)

Observe:

(f)

(g)

Observe:

(h)

Problems

Perform the indicated operations

(1) (6)

(2) (7)

(3) (8)

(4) (9)

(5) (10)

Answers

(1) (6)

(2) (7)

(3) (8)

(4) (9)

(5) (10)

Division of Algebraic Expressions

(1) Division of Two Monomials

Follow the laws of signed numbers and the laws of exponents

Examples:

(a)

(b)

(2) Division of a Polynomial by a Monomial

Divide each term of the polynomial by the monomial

Example:

(3) Division of One Polynomial by Another

(1) Arrange the dividend and divisor in descending powers of the variable

(2) Divide the first term of the dividend by the first term of the divisor. You get the first term of the quotient

(3) Multiply the divisor by the first term of the quotient and subtract this product from the dividend

(4) Repeat the process beginning with step (2)

Examples:

(a)

(b)

Problems

Divide

(1) (4)

(2) (5)

(3) (6)

Answers

(1) (4)

(2) (5) x - 4

(3) (6)