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Algebra and Trigonometry
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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, . . . .
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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, . . .
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EXAMPLE 9
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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
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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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EXAMPLE 10
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p = x2 5; q=x + 5 PolynomialGCD[p, q] 1 PolynomialGCD[p, q, Extension Automatic] 5+x PolynomialLCM[p, q] ( 5 + x) ( 5 + x2 ) PolynomialLCM[p, q, Extension Automatic] 5 + x2
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Although Mathematica will automatically expand integer exponents of products and quotients, if the exponent is non-integer, the expression will be left unexpanded. To force the distribution of the exponent, the command PowerExpand is available.
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PowerExpand[expression] expands nested powers, powers of products and quotients, roots of products and quotients, and their logarithms.
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EXAMPLE 11
(a b)5 a5 b5 (a b)
Mathematica distributes the exponent because it is an integer. Mathematica does nothing because the exponent is unde ned.
(a b)x PowerExpand[(a b) ] ax bx
We force the expansion with PowerExpand.
Algebra and Trigonometry
One must be very careful with PowerExpand when multi-valued functions are involved.
EXAMPLE 12
a b /. {a 1, b 1} 1 PowerExpand a b /. {a 1, b 1} 1
( 1)( 1) = 1 = 1
PowerExpand expands and then replaces the values of a and b by 1.
Here are a few additional examples illustrating PowerExpand:
EXAMPLE 13
(ax)y // PowerExpand ax y (a/b)x // PowerExpand ax b x Log[x y] // PowerExpand Log[x] + Log[y] Log[x/y] // PowerExpand Log[x] Log[y] Log[xy] // PowerExpand y Log[x]
SOLVED PROBLEMS
7.1 Test to see if 1 + x sin y + x 2 cos y + x 5e y is a polynomial in x. Is it a polynomial in y
SOLUTION
PolynomialQ[1 + x Sin[y] + x2 Cos[y] + x5 Exp[y], x] True PolynomialQ[1 + x Sin[y] + x2 Cos[y] + x5 Exp[y], y] False
y is treated as a constant in this expression.
7.2 What are the coefficients of the polynomial expansion of (2 x + 3)5
SOLUTION
poly = (2 x + 3)5; CoefficientList[poly, x] {243, 810, 1080, 720, 240, 32}
7.3 What is the coefficient of x y2 z3 in the expansion of (x + y + z)6
SOLUTION
poly = (x + y + z)3; Coefficient[poly, x y2 z3] 60
Algebra and Trigonometry
7.4 Expand (x + a + 1)4 completely.
SOLUTION
Expand[(x + a + 1)4] 1 + 4 a + 6 a2 + 4 a3 + a4 + 4 x + 12 a x + 12 a2 x + 4 a3 x + 6 x2 + 12 a x2 + 6 a2 x2 + 4 x3 + 4 a x3 + x4
7.5 Express (x + a + 1)4 as a polynomial in x.
SOLUTION
Collect[(x + a + 1)4, x] 1 + 4a + 6a2 + 4a3 + a4 + (4 + 12a + 12a2 + 4a3) x +(6 + 12a + 6a2) x2 +(4 + 4 a)x3 + x4
7.6 Factor the polynomial poly = 6 x3 + x2 y 11 x y2 6 y3 5 x2 z + 11 x y z + 11 y2 z 2 x z2 6 y z2 + z3 and solve for z so that poly = 0.
SOLUTION
poly = 6 x3 + x2 y 11 x y2 6 y3 5 x2 z + 11 x y z + 11 y2 z 2 x z2 6 y z2 + z3; Factor[poly]
(x + y z) (3 x + 2 y z) (2 x 3 y + z)
SOLUTION using Solve
Solve[poly 0,z] {{z x + y}, {z 3 x + 2 y}, {z 2 x + 3 y}}
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