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poly1 =(x + 1)10; poly2 = x3 5 x2 y + 3 x y2 7 y3; Variables[poly2] {x, y} Coef cient[poly1, x, 5] 252 Coef cient[poly2, x] 3 y2 Coef cient[poly2, y, 2] 3x Coef cient[poly2, x y2] 3 Coef cientList[poly1, x] {1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1}

2 2 2

- 197 -

Coef cientList[poly2, x] { 7 y3, 3 y2, 5 y, 1} Coef cientList[poly2, y] {x3, 5 x2, 3 x, 7}

Often it is convenient to write the solution of a polynomial equation as a logical expression. For example, if x2 4 = 0, then x = 2 or x = 2. Roots of polynomial equations can be expressed in this form using two specialized commands, Roots and NRoots. The solutions are given in disjunctive form separated by the symbol | | (logical or).

Roots[lhs rhs, variable] produces the solutions of a polynomial equation. NRoots[lhs rhs, variable] produces numerical approximations of the solutions of a polynomial equation.

M!1 0

solutions = Roots[x4 + x3 8 x2 5 x + 15 0, x] x 1( 1 13)| x 1( 1 + 13)| x 5| x 5 | | | 2 2 solutions && x > 2 //Simplify

Display routing table information. Display memory allocation and usage (mbufs).

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x 5 numericalsolutions = NRoots[x4 + x3 8 x2 5 x + 15 0, x] x 2.30278| 2.23607| 1.30278| 2.23607 |x |x |x numericalsolutions && x > 2 //Simplify x 2.23607

The division algorithm for polynomials guarantees that given two polynomials, p and s, for which degree(p) degree(s), there exist uniquely determined polynomials, q and r, such that p( x ) = q( x ) s( x ) + r ( x ) , where deg(r ) < deg(s)

We have x4 4 x4 4x2 4 4x2 x2 2 2 2x 2 x2 2 2x x2 2 2x : According to the method of partial fractions, assume 1 Ax B Cx D x4 4 x2 2x 2 x2 2x 2 Then 1 A C x3 B 2A 2C D x2 2A 2B 2C 2D x 2B 2D so that A C 0, B 2A 2C D 0, 2A 2B 2C 2D 0, 2B 2D 1 Solving simultaneously, A 1, B 1, C 1, D 1. Thus 8 4 8 4 dx 1 x 2 1 x 2 dx dx 8 x2 2x 2 x4 4 8 x2 2x 2 1 x 1 1 dx 1 x 1 1 dx dx dx 2 2 2 8 x 1 1 8 x 1 1 8 x 1 1 8 x 1 2 1 1 1 1 1 ln x2 2x 2 tan 1 x 1 ln x2 2x 2 tan 1 x 1 C 16 8 16 8 Then M

PolynomialQuotient[p, s, x] gives the quotient upon division of p by s expressed as a function of x. Any remainder is ignored. PolynomialRemainder[p, s, x] returns the remainder when p is divided by s. The degree of the remainder is less than the degree of s.

p = x5 7 x4 + 3 x2 5 x + 9; s = x2 + 1; q = PolynomialQuotient[p, s, x] 10 x 7 x2 + x3 r = PolynomialRemainder[p, s, x] 1 4 x

Example 15.3 netstat Display Routing $ netstat -rn Routing tables Destination Gateway.Flags Netmasks: (0) 0 ff00 (0) 0 ffff ff00

Expand[poly] expands products and powers, writing poly as a sum of individual terms. Factor[poly] attempts to factor poly over the integers. If factoring is unsuccessful, poly is unchanged. FactorTerms[poly] factors out common constants that appear in the terms of poly. FactorTerms[poly, var] factors out any common monomials containing variables other than var. Collect[poly, var] takes a polynomial having two or more variables and expresses it as a polynomial in var.

M!1 0

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