Polynomials of 2 Variables: General Form & Matrix Representation

  • Thread starter complexhuman
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In summary, the general form of a polynomial of 2 variables is shown as f(x,y) = a0 + a1x + a2y + a3x2 + a4xy + a5y2 + ... + anxmyn, where a0, a1, a2, ..., an are coefficients and m and n are non-negative integers. A polynomial of 2 variables can be represented in matrix form using a coefficient matrix. The degree of a polynomial of 2 variables is determined by the exponents in the last term, and it cannot have negative exponents. To add or subtract polynomials of 2 variables, simply combine like terms by adding or subtracting their coefficients.
  • #1
complexhuman
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how does one look like?I mean what's the general form? e.g. for a 1 var poly...general form = a0+a1x+a2^2+...+anx^n

and how could I represent that by a matrix?


Thanks
 
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  • #2
How about

[tex]a_0 x^n + a_1 x^{n-1} y + a_3 x^{n-2} y^2 + \cdot \cdot \cdot + a_n y^n + P(x) + Q(y)[/tex]

where P and Q are polynomials in their respective arguments?
 
Last edited:
  • #3
respective arguments?
 
  • #4
Arguments - meaning P(x) is a polynomial in x and Q(y) is a polynomial in y.
 

What is the general form of a polynomial of 2 variables?

The general form of a polynomial of 2 variables is shown as f(x,y) = a0 + a1x + a2y + a3x2 + a4xy + a5y2 + ... + anxmyn, where a0, a1, a2, ..., an are coefficients and m and n are non-negative integers.

How do you represent a polynomial of 2 variables in matrix form?

A polynomial of 2 variables can be represented in matrix form by using a coefficient matrix. The coefficient matrix is a 2-dimensional array with the coefficients of each term in the polynomial. For example, the polynomial f(x,y) = 2 + 3x + 4y + 5x2 + 6xy + 7y2 can be represented as [2 3 4; 0 5 6; 0 0 7].

What is the degree of a polynomial of 2 variables?

The degree of a polynomial of 2 variables is the highest power of both variables in the polynomial. In the general form, the degree is determined by the exponents in the last term of the polynomial. For example, the polynomial f(x,y) = 2 + 3x + 4y + 5x2 + 6xy + 7y2 has a degree of 2.

Can a polynomial of 2 variables have negative exponents?

No, a polynomial of 2 variables cannot have negative exponents. The exponents in a polynomial must be non-negative integers. If a polynomial has negative exponents, it is not considered a polynomial, but rather a rational function.

How do you add or subtract polynomials of 2 variables?

To add or subtract polynomials of 2 variables, you simply combine like terms. Like terms are terms that have the same variables and exponents. For example, to add f(x,y) = 3x2 + 4xy + 2y2 and g(x,y) = 2x2 + 3xy + 5y2, you would combine the coefficients of like terms to get h(x,y) = 5x2 + 7xy + 7y2.

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