Fundamental Concepts of Algebra
Constants: A symbol having a fixed numerical value is called a constant.
Variables or Literals: A symbol which takes on various numerical values is known as a variable or a literal.
, 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 the terms
Various Types of Algebraic Expressions
Monomial: An algebraic expression containing one term only, is called a monomial.
Binomial: An algebraic expression containing 2 terms is called a binomial.
Example:
(contains two terms
)
(contains two terms
)
Trinomial: An algebraic expression containing 3 terms is called a Trinomial.
Example:
(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.
Example:
(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.
Example:
(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.
Example:
, 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.
Example:
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.
Example:
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.
Example:
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).
Example: 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
.