Egwald's popular web pages are provided without cost to users. Please show your support by joining Egwald Web Services as a Facebook Fan:
Follow Elmer Wiens on Twitter:

where the coefficients c_{1}, c_{2}, ..., c_{n+1} are given real or complex numbers, and the variable x takes on takes on real or complex number values. A single term of the form c_{k} * x^{k+1} is called a monomial.

The degree of the polynomial p(x) is n if c_{n+1} is not zero. The nonzero coefficient with the highest power of x is the leading coefficient.

For example, the polynomial:

p(x) = 25 - 25 * x - 1 * x^{2} + 1 * x^{3}

is of degree 3, with 4 real number coefficients. If I evaluate the polynomial p(x) at x = 4:

The graph of the polynomial p(x), between -6 and 6, is displyed below.

The values of x at which p(x) takes on the value of zero, i.e. where the graph of p(x) crosses the x-axis, are called the roots of the polynomial p(x). The roots of p(x) are -5, 1, and 5.

Constant Polynomial (Functions).

A constant polynomial is of degree 0 since all terms but c_{1} are zero. For example, the polynomial:

c(x) = c_{1} = 6

is a constant polynomial. By convention, the zero polynomial with c_{1} = 0 has no degree.

Linear Polynomial (Function).

Linear polynomials have the form l(x) = c_{1} + c_{2} * x, and are of degree 1. For example, the polynomial:

l(x) = c_{1} + c_{2}* x = - 2 + 6 * x

is a linear polynomial.

The fundamental theorem of algebra states that any polynomial p(x) of degree n can be factored into the product of n linear polynomials, some of whose coefficients may be complex numbers.

Quadratic Polynomial (Function).

Quadratic polynomials have the form q(x) = c_{1} + c_{2} * x + c_{3} * x^{2}. For example, the polynomial:

Any polynomial of degree n with real coefficients can be factored into the product of quadratic polynomials with real coefficients, times one linear polynomial with real coefficients if n is an odd number.

The graph of an arbitrary quadratic can be obtained by setting the coefficients in the form that follows, and clicking "Graph".

Cubic Polynomial (Function).

Cubic polynomials have the form p(x) = c_{1} + c_{2} * x + c_{3} * x^{2} + c_{4} * x^{3}. For example, the polynomial:

The derivative of a polynomial is a polynomial of degree 1 less than the degree of the original polynomial.

Polynomials

Roots

Factors

p(x) = 25 - 25 * x - 1 * x^{2} + 1 * x^{3}

x_{1} = -5, x_{2} = 1, x_{3} = 5

-1*(-5 - x), (1 - x), (5 - x)

q(x) = p'(x) = -25 - 2 * 1 * x + 3 * 1 * x^{2}

x_{1} = -2.5726, x_{2} = 3.2393

3*(-2.5726 - x), (3.2393 - x)

l(x) = q'(x) = -2 + 2 * 3 * x = -2 + 6 * x

x_{1} = 1/3

-6*(1/3 - x)

c(x) = l'(x) = 6

c'(x) = 0

If one examines the graphs of the polynomials in the preceeding diagram, one sees that the graph of the derivative of a polynomial descends through the x-axis at that value of x where the original polynomial has a (local) maximum; it ascends through the x-axis at that value of x where the original polynomial has a (local) minimum.

Rolle's Theorem.

Between two consecutive real roots x_{1} and x_{2} of a polynomial p(x), its derivative polynomial p'(x) has at least one, or a greater odd number of roots. Furthermore, between two consecutive real roots x_{3} and x_{4} of p'(x), p(x) has at most one root.

Adding Polynomials.

Given two polynomials, p(x) of degree n, and f(x) of degree m, where without loss of generality (W.L.O.G.) m < n:

where q(x) = -1/9 + 1/3 * x, and r(x) = 200/9 - 152/9 * x.

Dividing polynomials by hand is a pain. With the form below, you can compute p(x) / f(x) for p(x) and f(x) up to degree 6. Just set their coefficients, and click "Divide". Be sure that the degree of p(x) >= degree of f(x). Leading zero coefficients are ignored.

p(x) = f(x)*q(x) + r(x)

q(x) =

16 * x^{0} + 11 * x^{1} + 4 * x^{2} + 1 * x^{3}

r(x) =

-17 * x^{0} + -75 * x^{1}

The Remainder Theorem.

The remainder after p(x) is divided by (a - x) is p(a):

p(x) = (a - x)*q(x) + r(x), implies

p(a) = (a - a)*q(x) + r(a) = r(a)

Polynomial Roots.

If x_{1} is a root of the polynomial p(x) of degree n, ie p(x_{1}) = 0, then by the remainder theorem, p(x) is divisible by (x_{1} - x):

p(x) = (x_{1} - x) p_{1}(x)

with p_{1} of degree n-1.

If x_{2} is another root of p(x) different from x_{1}, then

p(x_{2}) = (x_{1} - x_{2}) p_{1}(x_{2}) = 0

implies that

p_{1}(x_{2}) = 0

Hence, p_{1}(x) is divisible by (x_{2} - x). Therefore,

This procedure can be repeated, factoring the polynomial p(x) into the product of linear and/or quadratic polynomials.

The last equation provides some interesting information. The product of the roots of a quadratic equation equals the c_{1} coefficient; and, their sum equals the negative of the c_{2} coefficient. Furthermore, if a quadratic with real coefficients has a complex root, x_{1} = a + b*i, then its other root must be x_{2} = a - b*i, the complex conjugate of x_{1}. Then, x_{1}*x_{2} = a^{2} + b^{2}; and, x_{1} + x_{2} = 2*a.

search for real numbers c_{1} and c_{2} so that r(x) = 0. Then two roots of p(x) are:

x_{1} = (-c_{2} + sqrt(c_{2}^{2} - 4*c_{1})) / 2,

x_{2} = (-c_{2} - sqrt(c_{2}^{2} - 4*c_{1})) / 2.

After dividing p(x) by the quadratic q(x) = c_{1} + c_{2} * x + 1* x^{2}, repeat this process with p_{1}(x), etc., until all the roots of p(x) are obtained.

I use the procedure described above to find the roots of an arbitrary polynomial of up to degree 6. Enter the polynomial's coefficients in the form below and click "Roots". The new polynomial's roots and graph will display.

Roots of p(x)

root 1

root 2

root 3

root 4

root 5

-0

4

2 - 3*i

2 + 3*i

-3

Alternate Representations of Polynomials.

Polynomials can be expressed in the following forms

Chebyshev Polynomials can be used to approximate an arbitrary function f(x). Easy to perform operations, such as integration or differentiation, on the Chebyshev Polynomials replace operations on the function f(x).

The first six Chebyshev Polynomials, and their general forms are:

T_{0}(x) = 1

T_{1}(x) = x

T_{2}(x) = 2*x^{2} - 1

T_{3}(x) = 4*x^{3} - 3*x

T_{4}(x) = 8*x^{4} - 8*x^{2} + 1

T_{5}(x) = 16*x^{5} - 20*x^{3} + 5*x

T_{6}(x) = 32*x^{6} - 48*x^{4} + 18*x^{2} - 1

. . . . . . . . . .

T_{n+1}(x) = 2*x*T_{n}(x) - T_{n-1}(x)

T_{n}(x) = cos(n * arcos(x))

Notice that in the following diagram all roots of each T_{n}(x) polynomial occur in the [-1,1] interval of the x-axis. This holds true for each positive interger n. Furthermore, the range of each T_{n}(x) is [-1, 1] over the [-1, 1] domain.

References.

Ayres, Frank Jr. Modern Algebra. New York: Schaum McGraw-Hill, 1965.

Burden, Richard L. and J. Douglas Faires. Numerical Analyis. 6th ed. Pacific Grove: Brooks/Cole, 1997.

Miller, Norman, and Robert E. K. Rourke. An Advanced Course in Algebra. Toronto: MacMillan, 1941.

Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge UP, 1989.

Uspensky, J. V. Theory of Equations. New York: McGraw-Hill, 1948.