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EXAMPLE 3 Find all the solutions of x4 + x3 8x2 5x + 15 = 0 that are greater than 2.
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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
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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
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&& is Mathematica s logical and. See Section 7.4 for a discussion of Simplify.
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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)
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The Mathematica commands that produce the quotient and remainder are
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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.
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EXAMPLE 4
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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
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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.
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Algebra and Trigonometry
EXAMPLE 5
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)
EXAMPLE 6
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.
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
Powers of y are factored out. Powers of x are factored out.
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]
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.
EXAMPLE 8
Factor[x8 41 x4 + 400] ( 2 + x)(2 + x)( 5 + x2)(4 + x2)(5 + x2) Factor[x8 41 x4 + 400, GaussianIntegers True]
( 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) )
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