Fundamental Concepts of Algebra

Constants: A symbol having a __fixed numerical value__ is called a constant.

Example etc.

Variables or Literals: A symbol which takes on __various numerical values__ is known as a variable or a literal.

Example: , the value of will change with the value of

Algebraic Expressions: A __combination__ of constants and variables, __connected__ by operations is known as an algebraic expression.

Terms of an Algebraic Expression: Several parts of an algebraic expression separated by signs are called the terms of the expression.

Example:

(contains two terms )

(contains thee terms

Various Types of Algebraic Expressions

Monomial: An algebraic expression __containing one term only__, is called a monomial.

Example:

Binomial: An algebraic expression __containing 2 terms__ is called a binomial.

Examples:

(contains two terms )

(contains two terms )

Trinomial: An algebraic expression __containing 3 terms__ is called a Trinomial.

Examples:

(contains three terms )

(contains three terms )

Multinomial: An algebraic expression __containing more than 1 term__ is called a multinomial. Therefore, binomials and trinomials are also multinomial.

Examples:

(contains three terms )

(contains three terms )

Factors of a Term

When two or more quantities (numbers and literals) are multiplied to form a term, then each one of these quantities is called a factor of the term.

A constant factor is called a __numerical factor__ while a variable factor is called a __literal factor__.

Constant Term: In an algebraic expression, a term which has __no literal factor__ is called the constant term.

Examples:

(constant terms )

(constant terms )

Coefficients: Any factor of a term is called the coefficient of the product of all the remaining factors.

- In a term, the numeric factor attached to the variables is called, the
__numerical coefficient__. - In a term, the non-numeric factor consisting of the variables of the term is called the
__literal coefficient__.

Examples:

, the numerical coefficient is and the literal coefficient is . Also the coefficient of is and the coefficient of

Like Terms: Terms having the __same literal coefficients__ are called like terms otherwise they are called unlike terms.

Examples:

are like terms while are unlike terms.

Polynomials

An algebraic expression in which the variables involved have only __non-negative integral powers__, is called a polynomial. No variable should have negative power or be in the denominator in any expression.

Examples:

is a polynomial

is a polynomial

Degree of a Polynomial in One Variable: The __highest power__ of the variable in a polynomial of one variable is called the degree of the polynomial.

Examples:

is a polynomial of degree because the highest power on

is a polynomial of degree because the highest power on

is not a polynomial because in the term has a negative power of .

Linear Polynomial: Any polynomial of __degree 1__ is called a linear polynomial.

Example

is a linear polynomial of degree .

Quadratic Polynomial: Any polynomial of __degree 2__ is called a quadratic polynomial.

Example

is a quadratic polynomial of degree since the highest power on .

Cubic Polynomial: Any polynomial of __degree 3__ is called a cubic polynomial.

Example:

is a cubic polynomial of degree since the highest power on .

Constant Polynomial: A polynomial having __one term consisting of a constant__ is called a constant polynomial. Basically, you only have a constant in the polynomial. The degree of a constant polynomial is 0 (zero).

Examples: are all constant polynomials. Their degree is .

Degree of a Polynomial in Two or More Variables: If a polynomial involves two or more variables, then the __sum of the powers__ of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial in two or more variables.

Example:

is a polynomial in

Substitution: In any given expression, the __process of replacing or substituting__ each variable by one of its given values, is called substitution.

Example:

is a linear polynomial of degree . If , then the value of the polynomial is 16.