polynomials

An algebraic expression of the form a + bx + cx^2 + dx^3 + ………….. in which a, b, c, d are constants (numbers) and x is a variable, is called a polynomial in x.
The numbers a, b, c, d…….. are also called coefficients (therefore constants as they are numbers that do not change like variable x)

現在將 term with the highest power 寫先,

For quadratic expression, ax^2 + bx + c
For cubic expression, ax^3 + bx^2 + cx + d
For quartic expression, ax^4 + bx^3 + cx^2 + dx + e

When the variable x is raised to an index p to give x^p, the result is a power of x
Power refers to the number of times x is multiplied. For example, x^4 is the fourth power of x, x is raised to the power of 4..

Say a cubic expression, 2x^3 + 5x^2 + 6x + 7
The power of the first term is 3 (2x^3)
The power of the second term is 2 (5x^2)
The power of the third term is 1 (6x^1)
The power of the fourth term is 0 (7x^0), x^0 = 1,

The degree of a polynomial is the highest power of the variable.
For quadratic expression, ax^2 + bx + c , the degree is 2
For cubic expression, ax^3 + bx^2 + cx + d the degree is 3
For quartic expression, ax^4 + bx^3 + cx^2 + dx + e, the degree is 4

In secondary school, we don’t do polynomial of several variables.
Degree of a polynomial in two variables:
The degree of a polynomial which contains two or more variables is the greatest of the different sums of the powers of the variables in the various terms of the polynomial
Examples of degree of a polynomial in two or more variables:
In 2x^2y + 3xy + 4, the sum of the powers of variables x and y in:
2x^2y is (2 + 1) = 3,
3xy is (1 +1) = 2,
0 in 4.
Among the various sums of the powers of the variables in the terms, 3 is the greatest and therefore the degree of the above polynomial is taken as 3.

2014-07-02 04:03:20 補充：
Correction:
In secondary school, we don’t do polynomial equation of several variables, especially solving it.

Power is the highest index of the polynomial.
Eg. X^3y^2+ x^1y^2 , power =3 since the largest index is 3.

Degree is the largest sum of powers of a term in polynomials, that is in the above example

First term x^3y^2 has degree 3+2=5
Second term 1+2= 3
5 >3
Degree is 5