Degree of a Polynomial-
The degree of a polynomial in one variable is the highest index of powers of the variable in any term of the polynomial.
Example.1) The degree of the polynomial a⁴ + 3a² + 4 in the variable ‘a’ is 4 since this is the highest index of the powers of ‘a’ in the terms
Example.2) Similarly, the degree of the polynomial 5 x⁸ + 12 x⁵ - 8 x² is 8 since the highest of the indices of power of x is 8.
Example.3) Similarly, the degree of the polynomial 10 z⁶ - 12 z⁴ + 4 zᶟ is 8 since the highest of the indices of power of x is 8.
The degree of a polynomial in more than one variable is the highest of the sums of the indices of power of the variables in any of the terms of the polynomial.
Example.1) The expression 8 x⁶y⁴z + 7 xy²z⁵ - 10 x²y⁵z² is a polynomial in x, y & z. The degree of it’s terms are 6, 5, & 5 respectively. So the degree of the polynomial is ( 6 + 1 + 2 ) = 9
Polynomials are called by particular names in accordance with their degrees,
A) A polynomial of degree 1 is to be considered as a linear polynomial.
Example.1) x + 8 is a linear polynomial in one variable ‘x’
Example.2) - 4x + 8y is a linear polynomial in two variables ( a & b )
Example.3) 7a + 8b – 5c is a linear polynomial in three variables (x, y & z)
B) A polynomial of degree 2 is to be considered as a quadratic polynomial.
Example.1) 5 a², 16a² + 4 a, 8a² + 5 are quadratic polynomials in one variable.
Example.2) 8 a²b + 7b, 6a²b² + 4 ab, 8a²b² + 5ab² are quadratic polynomials in two variables.
Example.3) 8 a²bc + 7ab + 10 a, 6a²b² + 4 abc², and 4a²b²c + 3ab²c are quadratic polynomials in three variables.
C) A polynomial of degree 3 is to be considered as a cubic polynomial.
Example.1) 5aᶟ - 3a² + 2, - 3 xᶟ + 2 x, and 5 y² - 2 yᶟ + 8 are cubic polynomials in one variable.
Example.2) 8aᶟb² - 3a²b + 2, - 3 xᶟy - 5 xy², and 3 xy² - 6 xyᶟ + 12 are cubic polynomials in two variables.
Examples.3) 3xᶟy + 5 x²yzᶟ - 6 xᶟy, 6x²yᶟ - 7 xz, and 8x²zᶟ - 9xzᶟ are cubic polynomials in three variables.
D) A polynomial of degree 3 is to be considered as a quartic polynomial.
Example.1) 3x⁴ - 3aᶟ + 2x, - 8 xᶟ - 6 x⁴, and 12 y⁴ - 2 yᶟ + 8 are quartic polynomials in one variable.
Example.2) 10aᶟb⁴ - 3a²b + 8, - 6 xᶟy⁴ + 8 xᶟy², and 3 xy²- 7 x⁴y + 12 are quartic polynomials in two variables.
Examples.3) 3xᶟyz⁴ + 5 yzᶟ - 8 xᶟy, 6x²yᶟ - 3 xy⁴z, and 8x²yz⁴ - 9xz are quartic polynomials in three variables.
Examples.4) 8wxᶟyz⁴ + 6 xyzᶟ - 8 xᶟw, 6 wx⁴yᶟ - 3 xy⁴z, and 8 wx²yz⁴ - 9x²z are quartic polynomials in four variables.