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Roots[poly 0, z] z x + y||z 3 x + 2 y||z 2 x + 3 y
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7.7 Find the quotient and remainder when x5 + 2 x4 3 x3 + 7x2 10 x + 5 is divided by x2 4 and verify that the answer is correct.
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p = x5 + 2 x4 3 x3 + 7 x2 10 x + 5; s = x2 4; q = PolynomialQuotient[p, s, x] 15 + x + 2 x2 + x3 r = PolynomialRemainder[p, s, x] 65 6 x checkpoly = q * s + r//Expand 5 10 x + 7 x2 3 x3 + 2 x4 + x5 checkpoly p True
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7.8 Express (x + y + z)3 as a polynomial in z.
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Collect[(x + y + z)3, z] x3 + 3 x2 y + 3 x y2 + y3 + (3 x2 + 6 x y + 3 y2)z + (3 x + 3 y)z2 + z3
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7.9 Let p = 2x4 15x3 + 39x2 40x + 12 and q = 4x4 24x3 + 45x2 29x + 6. Compute their GCD and LCM and show that their product is equal to pq.
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p = 2 x4 15 x3 + 39 x2 40 x + 12; q = 4 x4 24 x3 + 45 x2 29 x + 6; a = PolynomialGCD[p, q] 6 + 17 x 11 x2 + 2 x3 b = PolynomialLCM[p, q] ( 2 + x)(6 29 x + 45 x2 24 x3 + 4 x4) Expand[a * b] Expand[p * q] True
7.10 Factor x 4 25 over the integers and then over the field containing 5 and i.
SOLUTION
Factor[x4 25] ( 5 + x2)(5 + x2) Factor[x4 25, Extension { 5 , I}] ( 5 x)( 5 x)( 5 + x)( 5 + x)
7.11 Expand ln
SOLUTION
x a yb . c z
xa y b / /PowerExpand Log zc 1 (a Log[x]+ b Log[y] c Log[z]) + 2
7.2 Rational and Algebraic Functions
There are a few commands appropriate for use with rational functions (fractions).
Numerator[ fraction] returns the numerator of fraction. Denominator[ fraction] returns the denominator of fraction. Cancel[ fraction] cancels out common factors in the numerator and denominator of fraction. The option Extension Automatic allows operations to be performed on algebraic numbers that appear in fraction. Together[ expression] combines the terms of expression using a common denominator. Any common factors in numerator and denominator are cancelled. Apart[ fraction] writes fraction as a sum of partial fractions.
EXAMPLE 14
x2 + 5 x + 6 Cancel 2 x + 3 x + 2 3+ x 1+ x
EXAMPLE 15
Together 1 + 2 2 x +1 x 1 1 1 + x
Algebra and Trigonometry
EXAMPLE 16
2 Apart 4 x + 5 x x + x3 x 1
1 + 2 + 1 3 x 1 + x 1 + x 1 + x + x2
Since Mathematica, by default, converts factors with negative exponents to their positive exponent equivalents, the result of Numerator or Denominator may be different than expected.
EXAMPLE 17
fraction =
x 1 y 2 ; z 3 Numerator[fraction] z3 Denominator[fraction] x y2
ExpandNumerator[expression] expands the numerator of expression but leaves the denominator alone. ExpandDenominator[expression] expands the denominator of expression but leaves the numerator alone. ExpandAll[expression]expands both numerator and denominator of expression, writing the result as a sum of fractions with a common denominator.
EXAMPLE 18
expression = (x + 1)(x + 2); (x + 3)(x + 4) ExpandNumerator[expression] 2 + 3 x + x2 (3 + x)(4+ x) ExpandDenominator[expression] (1 + x)(2 + x) 12 + 7 x + x2 ExpandAll[expression] 2 3x x2 + + 12 + 7 x + x2 12 + 7 x + x2 12 + 7 x + x2 ExpandNumerator[ExpandDenominator[expression]] 2 + 3 x + x2 12 + 7 x + x2
The commands described in this section are not limited to rational functions (quotients of polynomials) but will work for both algebraic expressions involving radicals and non-algebraic expressions involving functions or undefined objects. In addition, if the option Trig True is set within the command, Mathematica will use standard trigonometric identities to simplify the expression. This will be discussed further in Section 7.3.
EXAMPLE 19
6 Expand 1 + x
1 + 6 x + 15 x + 20 x3/2 + 15 x2 + 6 x5/2 + x3
Algebra and Trigonometry
EXAMPLE 20
1 Apart ( x + 1) ( x + 2) 1 1 1+ x 2+ x
SOLVED PROBLEMS
7.12 The expression
f ( x ) f (a) appears in calculus in connection with the derivative. Simplify this x a 9 expression for f(x) = x , a = 3.
SOLUTION
f[x_]= x9; a = 3;
Cancel f[x] f[a] x a
6561 2187 x + 729 x2 243 x3 + 81 x4 27 x5 + 9 x6 3 x7 + x8 a c e 7.13 Express the sum of , , and as a single fraction. b d f
SOLUTION
Together[a/b + c/d + e/f]
b d e+ b cf+a df bdf
7.14 Write
(x + 2)(x 2 + 3)(2 x 7) with expanded numerator and denominator. (x 2 + 5 x + 2)(x 5)(x + 6)
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