The concept of algebra is very useful in our day to day lives. Let us acquaint ourselves with the basic terms of algebra.
Constants and Variables
A constant is any numeral on the number line. It may be negative or positive. It is the number placed before multiplying the variable in an algebraic expression.
A variable on the other hand is an assumed value for a term which changes with the change in the constant term. For example, if we have an equation x + y = 5, there will be different values of x and y which satisfy this equation. If the value of a variable x is 1 then value of variable y will be 4. Likewise if the value of x is 2 then the value of y will be 3. Thus, the value of variables keeps changing.
A coefficient is a combination of a constant and a literal term. It can be of two kinds.
Constant coefficient – the constant coefficient is the numeric coefficient of a term. In our example of ‘8x’, 8 is the numeric coefficient.
Literal coefficient – all variables of a term are its literal coefficients. In the expression 8xy, ‘8’ is the numeric coefficient whereas ‘xy’ is the literal coefficient.
Like terms are terms in which the literal coefficient and their exponents are the same. In such terms only the constant coefficient can change. For example, terms 8xy and 3xy will be called like terms because their literal coefficient ’xy’ and their exponent 1 is same. The important thing about like terms in algebra is that the arithmetic operations of addition and subtraction can solely be performed on like terms.
Opposite to like terms, in unlike terms both the numeric as well as the literal coefficient is changed. For example, 3xy and 5xy2 are unlike terms since their literal coefficients, ‘xy’ and ‘xy2’ and their respective exponents are not same. The operations of addition and subtraction cannot be performed on unlike terms.
An algebraic expression is a phrase which contains numeric coefficients (constants), literal coefficients (variables) and arithmetic operators of either addition (+), subtraction (-), multiplication(x) and division(÷). For example, 5xy + 3x2y + 2y2 is an algebraic expression.
A polynomial is an expression which consists of constants and variables and uses the operands of addition, subtraction, multiplication and positive integer exponents.
For example, x is x2 − 5x + 9 is a polynomial in single variable, while x3 + 9xyz2 − yz + 6 is a polynomial in three variables. Based on the number of terms in an algebraic expression, they can be classified as monomials, binomials and trinomials.
Mono means one. Hence, a polynomial with a single algebraic term is called a monomial. 5x2y, 3xy, 2x2 are monomials.
Bi means two. Polynomials in which the number of algebraic terms are two are called binomials. Example: 5x+3y, 5x2-3y2 etc. Binomials are written with either an addition or a subtraction sign in between.
Tri means three. Polynomials with three different algebraic terms is called a trinomial. For example, 5x+3y2+4z is a trinomial. Trinomials too, like binomials are written either with an addition or a subtraction sign in between.
Degree of polynomials
The greatest exponent in a polynomial or algebraic expression is termed as its degree. In other words, the power of the literal coefficient in an algebraic expression is its degree. For example, in expression ‘6x’, the degree of the literal coefficient ‘x’ is one. Thus, it is a monomial of power one. Similarly, 6x2 has a power of two. In an expression where the literal coefficients are more than one, the degree of the expression is determined by adding the powers of all the literal coefficients. For example, in the expression ‘3x2y2’, the power will be 4 ie. 2+2.
In a polynomial, the highest power of an algebraic expression is said to be the degree of the polynomial. For instance, 3x2 + 4x2y + 5x3y + 7xy6 is a polynomial with four different algebraic expressions. The power of these expressions is 2, 3, 4 and 7 respectively. Thus, the highest power, i.e. 7 will be the degree of this polynomial.