Polynomials in several variables
While we have used polynomials such as x2+y2 ,we have never actually considered them as functions of two variables.This is a concept we need for the study of graphs and other figures in space.A function of two variables f(x,y) is a rule that assigns a real number to each point (x,y) in some subset D of the plane.The set D is called the domain of the function.
Examples:
(1) f(x,y) = x+3y ,domain the whole plane.
(2) f(x,y) = arc sin(x-y) , domain all (x,y) with -1 x-y 1.
Our main concern is polynomial functions.In two variables a polynomial functions the sum of a finite number of terms (monomials) cxmyn,where c is a real and m and n are non-negative integers.The domain of any polynomial function of two variables is the whole plane.
Examples:
(1) Linear polynomial : a+bx+cy
(2) Quadratic polynomial : a+bx+cy+dx2+exy+fy2
(3) Cubic polynomial : (quadratic)+ax3+bx2y+cxy2+dy3
A function of three variables f(x,y,z) is a rule that assigns a real number to each point (x,y,z) in some subset D of 3-space. In three variables,a polynomial function is a sum of monomialscxmynzp;its domain is all of 3-space.
Examples: f(x,y,z) =x2+y2+z2. This polynomial function assigns to each point (x,y,z) the square of its distance from 0.
Algebra of polynomials
Two polynomials can be added, subtracted, or multiplied; the result is again a polynomial. For addition and subtraction this is obvious; for multiplication it follows from the rule:
(xmynzp)(xrys zt)=xm+ryn+szp+t
and the distributive laws.
Factoring
A polynomial can some times be factored into the product of two or more other polynomials.This has the same advantages as factoring of polynomials in one variable; it expresses the given function in terms of simpler ones and often simplifies algebraic and numerical computations.Factoring in several variables, however, can be difficult.
Examples :
x3+y3=(x+y)(x2-xy+y2),
x3+y3+z3-3xyz=(x+y+z)(x2+y 2+z2-xy-yz-xz),
x3+x2z+xy+yz=(x+z)(x2+y).
Degree
The degree of a non-zero monomial is defined by
deg(axmynzp)=m+n+p.
The degree of polynomial (sum of monomials) is the highest degree of its (non-zero) terms.The polynomial f(x,y,z)=0 is usually not assigned a degree.
Thus
deg(x3+yz)=3, deg(x3+z4)=4,
deg(x2yz+z3)=4, deg(x2y2z+3yz4)=5.
Adding two polynomials cannot produce a term of degree higher than the degrees of all the terms in either summand , hence either f+g=0,or deg(f+g)maximum(deg f,deg g),provided f and g are non-zero polynomials.
It can be shown that deg(fg)=deg f + deg g,but the proof is somewhat complicated.
Graphs
Let f(x,y) be a function of two variables.The set of zeros of f is the set of all points (x,y) in the domain for which f(x,y)=0. It is also called the graph(or locus) of the equation f(x,y)=0. Note that it is notcalled the graph of the function f.That is something quite different as we see shortly.
Examples:
(1) f(x,y)=ax+by+c.
The set of zeros of f(x,y),all points (x,y) where ax+by+c=0,is
a lineif a2+b2 > 0
the planeifa=b=c=0
emptyifa=b=0 , c ≠ 0 .
The graph of f(x,y), all points (x,y,z) where z=ax+by+c, is a plane in space no matter what a,b,c are.
(2) f(x,y)=1-x2.
The set of zeros consists of all points (x,y) where x2=1,that is ,two vertical lines (Fig. 1(a)).
Fig. 1f(x,y)=1-x2
The graph consists of all points (x,y,z) where z=1-x2. This equation z=1-x2 is free of y.Hence if (xo,yo,zo) is on the graph,so are all points (xo,yo,zo) for all values of y. These points form a line parallel to the y-axis. We say the graph is a cylinder with generators parallel to the y-axis.
To understand the shape of the cylinder, we note that the graph intersects the z,x-plane (y=0)</i> in the points (x,0,z),where z=1-x2. This curve is a parabola in the z,x-plane.Therefore the graph is a parabolic cylinder (Fig. 1(b)).