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Ebook Mathematica (2/E)1: Part 2

Part 2 book “Mathematica” has contents: Rational and algebraic functions, the art of simplification, differential calculus, integral calculus, multivariate calculus, maximum and minimum values, multiple integrals, ordinary differential equations, matrix operations, linear systems of equations, and other contents. | CHAPT E R 7 Algebra and Trigonometry 7.1 Polynomials Because they are so prevalent in algebra, Mathematica offers commands that are devoted exclusively to polynomials. PolynomialQ[expression, variable] yields True if expression is a polynomial in variable, and False otherwise. Variables[polynomial] gives a list of all independent variables in polynomial. Coefficient[polynomial, form] gives the coefficient of form in polynomial. Coefficient[polynomial, form, n] gives the coefficient of form to the nth power in polynomial. CoefficientList[polynomial, variable] gives a list of the coefficients of powers of variable in polynomial, starting with the 0th power. EXAMPLE 1 PolynomialQ[x2 + 3 x + 2, x] True PolynomialQ[x2 + 3 x + 2/x, x] False PolynomialQ[x2 + 3 x + 2/y, x] ← 2/y is treated as a constant with respect to x. True PolynomialQ[x2 + 3 x + 2/y, y] False EXAMPLE 2 poly1 =(x + 1)10; poly2 = x3 – 5 x2 y + 3 x y2 – 7 y3; Variables[poly2] {x, y} Coefficient[poly1, x, 5] 252 Coefficient[poly2, x] 3 y2 Coefficient[poly2, y, 2] 3x Coefficient[poly2, x y2] 3 CoefficientList[poly1, x] {1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1} 186 CHAPTER 7 Algebra and Trigonometry 187 CoefficientList[poly2, x] {–7 y3, 3 y2, –5 y, 1} CoefficientList[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. EXAMPLE 3 Find all the solutions of x4 + x3 – 8x2 – 5x + 15 = 0 that are greater than 2. solutions = Roots[x4 + x3 – 8 x2 – 5 x + 15 0, x] x 1(–1 – 13)|| x 1(–1 + .