Degree of a Term –
The highest index of power or the sum of the indices of power of the variable(s) in a term is called the degree of the term.
Examples.1) – 5 x⁴ is of the four degree
Examples.2) the degree of the term a⁴b²c⁵d⁶ is (4 + 2 + 5 + 6) = 17
Examples.3) the degree of the term 8 x⁴yᶟz⁵ is (4 + 3 + 5) = 12
Example.4) 15 aˉ⁴b⁶c⁸ is a term of degree is {(-4) + 6 + 8 } = 10
Polynomial in one Variable –
An algebraic expression is called a polynomial if it is a finite sum of terms that contain only non-negative integral exponents of a variable. A polynomial in one variable ( say x ) contains only such terms as can be expressed in the form axⁿ, where ‘a’ is a constant and ‘n’ is a non-negative integer. In other words, a polynomial in x cannot have terms, such as 1/x, 1/x², 1/xᶟ and 1/x⁴ because they are not of the form xⁿ, where ‘n’ is a non-negative integer.
Example.1) - 5x⁴ - 7 x + 3 is a polynomial in ‘x’ with three terms - 5x⁴, - 7x, and here the term 3 is called a constant term.
5x
Example.2) - x² + -------- + 7 is not a polynomial because the
3
5x
term -------- is not a form of axⁿ, where ‘n’ is a non-negative integer.
3
Polynomial in two or more Variable –
A polynomial in two or more variables is a sum of terms that contain only non-negative integral exponents of those variables. For example, a polynomial in x, y & z is a sum of one or more terms of the form n xᵐ yᵖ zᵏ, where ‘n’ is any constant and m, p, & k are non-negative integers.
Examples.1) 10y²z + 8 x²y + 3 xz² is a polynomial in three variables x, y, & z.
Example.2) 21a⁴bcd⁴ + 8 ab²c²d - 10 abc⁵dᶟ is a polynomial in four variables a, b, c, and d.