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poly = 6 x2 y3z4 + 8 x3 y2 z5 + 10 x2 y4 z3; Factor[poly] 2 x2 y2 z3(5 y2 + 3 y z + 4 x z2) FactorTerms[poly] 2(5 x2 y4 z3 + 3 x2 y3 z4 + 4 x3 y2 z5) FactorTerms[poly, x] 2 y2 z3(5 x2 y2 + 3 x2 y z + 4 x3 z2) FactorTerms[poly, y] 2 x2 z3(5 y4 + 3 y3 z + 4 x y2 z2) FactorTerms[poly, z] 2 x2 y2(5 y2 z3 + 3 y z4 + 4 x z5)

Only the common factors not involving z are factored. Only the common factors not involving y are factored. Only the common factors not involving x are factored. Only the constants are factored. poly is factored completely.

! ( ) dx 1 M 2 2M 2 1 1 tan 1 M 1 tan 1 M 1 lim ln 8 8 8 x4 4 M!1 16 M 2 2M 2 1 dx , called an improper integral of the rst kind. Such integrals are considered x4 4 0 See also Problem 5.74.

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poly = 1 + 2 x + 3 y + 4 x y + 5 x2 y + 6 x y2 + 7 x2 y2; Collect[poly, x] 1 + 3 y + x(2 + 4 y + 6 y2) + x2 (5 y + 7 y2) Collect[poly, y] 1 + 2 x +(3 + 4 x + 5 x2)y +(6 x + 7 x2)y2

Route Tree for Protocol Family 2: default 145.142.210.100 UG 127 127.0.0.1 U 140.142.78 145.142.210.20 U

EXAMPLE 7 The following Manipulate command expands (x + 1)n to any power between 1 and 10, controlled

by radio buttons. Manipulate[Expand[(x + 1)n] //TraditionalForm,{n, Range[10]}, ControlType RadioButton]

CHAP. 5]

By default, Factor allows factorization only over the integers. There are options that allow this default to be overridden.

Extension {extension1, extension2, . . .} can be used to specify a list of algebraic numbers that may be included as well. (The brackets, {}, are not needed if only one extension is used.) Extension Automatic extends the field to include any algebraic numbers that appear in the polynomial. GaussianIntegers True allows the factorization to take place over the set of integers with adjoined. Alternatively, or I may be included in the list of extensions.

22 3 4

Factor[x8 41 x4 + 400] ( 2 + x)(2 + x)( 5 + x2)(4 + x2)(5 + x2) Factor[x8 41 x4 + 400, GaussianIntegers True]

0 x!0

( 2 + x)( 2 + x)(2 + x)(2 + x)( 5 + x2)(5 + x2)

Factor[x8 41 x4 + 400, Extension 5 ] ( 5 x)( 2+ x)(2+ x)( 5 + x)(4+ x2)(5+ x2) Factor[x8 41 x4 + 400, Extension {I, 5 }] ( 5 x)( 5 x)( 5 + x)( 2+ x)( 2 +x)(2 + x)(2+ x)( 5 + x) )

The greatest common divisor (GCD) of polynomials, p1, p2, . . . is the polynomial of largest degree that can be divided evenly (remainder = 0) into p1, p2, . . . . The least common multiple (LCM) of polynomials p1, p2, . . . is the polynomial of smallest degree that can be divided evenly by p1, p2, . . . .

The conditions of L Hospital s rule are satis ed, so that the required limit is d x d sin t3 dt sin x3 sin x3 3x2 cos x3 1 dx 0 lim dx lim lim lim 3 d 4 x!0 x!0 4x x!0 d x!0 4 12x2 x 4x3 dx dx

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PolynomialGCD[p1, p2,...] computes the greatest common divisor of the polynomials p1, p2, . . . PolynomialLCM[p1, p2,...] computes the least common multiple of the polynomials p1, p2, . . .

p = (x 1)(x 2)2(x 3)3; q = (x 1)2(x 2)(x 3)4; PolynomialGCD[p, q] ( 3 + x)3( 2 + x)( 1 + x) PolynomialLCM[p, q] ( 3 + x)4( 2 + x)2( 1 + x)2

and mk representing the l.u.b. and g.l.b. of f x in the interval xk 1 ; xk , i.e., such that mk @ f x @ Mk . We then have m b a @ s

By default, both PolynomialGCD and PolynomialLCM assume the coefficients of the polynomials to be rational numbers. As with Factor, the option Extension can be used to specify a list of algebraic numbers (and/or I) that may be allowed.

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