5. Types of algebraic expressions. Polynomials

Polynomial
    A polynomial is an algebraic expression with one or more terms.
    For example : 6x, x³ + 3x² + 9x + 7, 3x² – 4xy + 7y² etc. are all polynomials.   

Degree of polynomial
    The degree of a polynomial of one variable is the highest power of the variable in the given polynomial. For example P(x) = 2x³ + 3x² – 6x + 4. The highest power of x in all terms of polynomial is 3. 
    Hence, the degree of the polynomial is 3.  
    We can classify polynomial according to its degree.
    (i)     Constant polynomial : Polynomial having degree zero is known as constant polynomial. 

(ii)     Linear polynomial : Polynomial having degree one is known as linear polynomial. 
        For ex : 2x – 5, x + 3.
(iii)   Quadratic polynomial : Polynomial having degree two is known as quadratic polynomial. 
        For ex :  x² + 1, 7x², x² + 2x – 1.
(iv)   Cubic polynomial : Polynomial having degree three is known as cubic polynomial. 
        For ex :  7x³ + 5x² + 1, x³ – x + 1.
(v)   Biquadratic polynomial : Polynomial having degree four is known as biquadratic polynomial. 
        For ex : x⁴ + 1, x⁴ + x² + 1.

Factors : Each term in an algebraic expression is a product of one or more number(s) and/or literal number(s).

These number(s) and/or literal number(s) are known as the factors of that term.
    (i)     In the binomial 8ab + 3c, 8ab and 3c are two terms. In the term 8ab, 8, a and b are its factors. Clearly, number 8 is the numerical factor, and a and b are literal factors.
    (ii)    In the binomial expression – ab – 5, the term – ab has – 1 as the numerical factor while a and b are literal factors.

Coefficient : In a term of an algebraic expression any of the factors with the sign of the term is called the coefficient of the product of the factors.
Consider the term – 5ab in the binomial – 5ab + 7. The coefficient of a in the term – 5ab is – 5b, the coefficient of b is – 5a and the coefficient of ab is – 5.

Types of algebraic expressions. Polynomials

Polynomial (poly - 'many', nomial - 'term').

 When an algebraic expression has one or many terms, then that expression is called a polynomial expression.
In some cases, polynomial multiplication can be performed easier when using the abridged multiplication formulas.

You need to remember 3 formulas:

1) The square of the sum of two expressions:
(a+b)²=a²+2ab+b²

2) The square of the difference of two expressions:
(a−b)²=a²−2ab+b²

3)  The difference of squares of two expressions:
(a−b)(a+b)=a²−b²