Fundamental Concepts of Algebra

Constants: A symbol having a fixed numerical value is called a constant.

$\displaystyle \text{Example: } 3,\ 5,\ \frac{2}{5} , \ -5, \sqrt{10} \text{ etc. }$

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

$\displaystyle \text{Example: } x=3y$ , the value of  $\displaystyle x$ will change with the value of  $\displaystyle y$

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  $\displaystyle x, \ +,\ or\ -$ signs are called the terms of the expression.

Example:

$\displaystyle 3x+7y$ (contains two terms  $\displaystyle 3x \ and\ 7y$ )

$\displaystyle 3x^2+2y^3-3xy$ (contains the terms  $\displaystyle 3x^2, 2y^3, \ and\ -3xy)$

Various Types of Algebraic Expressions

Monomial: An algebraic expression containing one term only, is called a monomial.

$\displaystyle \text{Example: } 3x, 5y, -2x, y^3, - \frac{4x}{9y}$

Binomial: An algebraic expression containing 2 terms is called a binomial.

Example:

$\displaystyle 3x+7y$ (contains two terms  $\displaystyle 3x \ and\ 7y$ )

$\displaystyle x^2y+2 \frac{z}{3}$ (contains two terms  $\displaystyle x^2y \ and \ 2 \frac{z}{3}$)

Trinomial: An algebraic expression containing 3 terms is called a Trinomial.

Example:

$\displaystyle 3x+7y+3z$ (contains three terms  $\displaystyle 3x, 7y, \ and\ 3z$ )

$\displaystyle x^2y+2 \frac{z}{3} -2abc$ (contains three terms  $\displaystyle x^2y, 2 \frac{z}{3} \ and\ -2abc$ )

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

Example:

$\displaystyle 3x+7y+3z$ (contains three terms  $\displaystyle 3x, 7y, \ and\ 3z$ )

$\displaystyle x^2y+2 \frac{z}{3} -2abc+c^2$ (contains three terms  $\displaystyle x^2y, 2 \frac{z}{3} -2abc\ and\ c^2$ )

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:

$\displaystyle 3x+7y+3z-2$ (constant terms  $\displaystyle -2$ )

$\displaystyle x^2y+2 \frac{z}{3} -2abc+c^2+21$ (constant terms  $\displaystyle 21$ )

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:

$\displaystyle -9xy$ , the numerical coefficient is  $\displaystyle -9$ and the literal coefficient is  $\displaystyle xy$ . Also the coefficient of  $\displaystyle x$ is  $\displaystyle -9y$ and the coefficient of  $\displaystyle y \ is\ -9x$

Like Terms: Terms having the same literal coefficients are called like terms otherwise they are called unlike terms.

Example:

$\displaystyle 7x^2, -7x^2, - \frac{3}{2} x^2$ are like terms while  $\displaystyle 7x^2y, -7z^2, - \frac{3}{2} xz^2$ 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:

$\displaystyle 3x+7y+3z-2$ is a polynomial

$\displaystyle x^2y+2 \frac{z}{3} -2abc+c^2+21$ 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:

$\displaystyle 3x+7x^2+3x^3-2$ is a polynomial of degree  $\displaystyle 3$ because the highest power on  $\displaystyle x\ is\ 3$

$\displaystyle 3y-4y^2+3y^5$ is a polynomial of degree  $\displaystyle 5$ because the highest power on  $\displaystyle y\ is\ 3$

$\displaystyle 3x+ \frac{7}{x}$ is not a polynomial because in the term  $\displaystyle \frac{7}{x} , \ x$ has a negative power of  $\displaystyle -1$ .

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

Example:

$\displaystyle 3x+7$ is a linear polynomial of degree  $\displaystyle 1$ .

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

Example:

$\displaystyle 7x^2+3x+1$ is a quadratic polynomial of degree  $\displaystyle 2$ since the highest power on  $\displaystyle x \ is\ 2$ .

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

Example:

$\displaystyle 7^3+3x+1$ is a cubic polynomial of degree  $\displaystyle 3$ since the highest power on  $\displaystyle x is 3$ .

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: $\displaystyle 7, 4, 45, -23, - \frac{2}{3}$ are all constant polynomials. Their degree is  $\displaystyle 0$ .

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:

$\displaystyle x^3y^2+x^2-2x^2y^4+ \frac{2}{3} x^1y^3z^4$ is a polynomial in  $\displaystyle x,\ y,\ and\ z\ of \ degree\ 8$

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

Example:

$\displaystyle 3x+7$ is a linear polynomial of degree  $\displaystyle 1$ . If  $\displaystyle x=3$ , then the value of the polynomial is  $\displaystyle 16$.