- #1
Jhenrique
- 685
- 4
If a polynomial of 1 variable, for example: P(x) = ax²+bx+c, can be written as P(x) = a(x-x1)(x-x2), so a polynomial of 2 variables like: Q(x,y) = ax²+bxy+cy²+dx+ey+f can be written of another form?
pasmith said:[tex]
(px + qy + r)(sx + ty + u) = psx^2 + (pt + qs)xy + qty^2 + (pu + rs)x + (qu + rt)y + ru
[/tex]
That gives you six equations in six unknowns.
There is no general solution, because you can pretty quickly eliminate [itex]s = a/p[/itex], [itex]t = c/q[/itex] and [itex]u = f/r[/itex] to end up with [tex]
cp^2 + aq^2 = bpq \\
fp^2 + ar^2 = dpr \\
fq^2 + cr^2 = eqr.[/tex] These are cylinders in [itex](p,q,r)[/itex] space whose cross-sections are conic sections in the [itex](p,q)[/itex], [itex](p,r)[/itex] and [itex](q,r)[/itex] planes respectively. There is no reason why these should all intersect (it's pretty easy to arrange three such cylinders of circular cross-section so that they don't intersect), and if they do all intersect they may do so at multiple points.
A polynomial of 2 variables is an algebraic expression that contains two variables, usually represented by x and y, and can be written as a sum of terms where each term is a constant multiplied by a power of the variables.
The degree of a polynomial of 2 variables is determined by the highest power of the variables that appears in any term of the expression. For example, the polynomial 3x^2y^3 has a degree of 5 (2+3).
A monomial is a polynomial with only one term, while a binomial is a polynomial with two terms. For example, 5x is a monomial and 3x^2 + 2x is a binomial.
Yes, a polynomial of 2 variables can have a negative exponent. This means that the variable is being raised to a negative power, which is equivalent to taking the reciprocal of the variable raised to the positive power. For example, x^-2 is equivalent to 1/x^2.
The main difference between a polynomial of 2 variables and a polynomial of 1 variable is the number of variables present. A polynomial of 1 variable only contains one variable, while a polynomial of 2 variables contains two variables. Additionally, the degree of a polynomial of 2 variables can be different for each variable, while the degree of a polynomial of 1 variable is the same for all terms.