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CHAPTER FOUR: LINEAR, QUADRATIC AND CUBIC EQUATIONS
- Hong Kong University Press, HKU
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CHAPTER FOUR LINEAR, QUADRATIC AND CUBIC EQUATIONS A polynomial g(x} = bmxm +bm_Ixm- 1 +... +bix +bo defines a polynomial function g(x} : R -+ R which maps every real number c of the domain to the real number g(c} of the range. The evaluation of g(x} at x = c is a very staight-forward matter and there are simple methods of calculation by which the correct value of g(c} can be obtained. We are now interested in the possibility of finding real values c of the domain such that g(c} coincides with an pre-assigned value d of the range. Thus given g(x} E R[x] and dE R, we seek information on the possible values of c such that g(c} = d. In the language of set theory , the problem is to find the pre-images c of d under the mapping g(x} : R-+ R. After absorbing the number -d into the constant term of g(x), i.e. replacing g(x) by f(x) = g(x) - d, this amounts to the evaluation of all real roots c of the polynomial function f(x) . In contrast to the evaluation of a polynomial !(x) at a given value of x, the problem of finding roots of a given polynomial function f(x) is a very difficult problem of mathematics. In this chapter, we shall study the methods of solving some simple equations. 4.1 Terminology Let f(x) = anxn +an_IXn- 1 +...+alX +ao be a polynomial in the indeterminate x with real coefficients. If we regard the symbol x in the above expression as a definite but unknown real or complex number , then the expression simply represents a number. Since numbers can be compared by equality, it is therefore legitimate to say that we wish (A) To find the values of the unknown number x such that anxn+ anxn- 1 + ... + alX + ao = o. 65 Polynomials and Equations This being the problem at hand, we may also say (A) in anyone of the following ways: (B) To solve for x in the polynomial equation (C) To find all roots of the equation Furthermore given a polynomial f(x) = anxn + an_1Xn- 1+ ... + alX + ao in the indeterminate x, we may also use the abbreviated expression f(x) = 0 for the equation in the unknown x. Terminology such as degree, coefficients, terms, etc. of an equation in the unknown x shall have the obvious meaning. Moreover a solution, a root and a zero of an equation f(x) = 0 all mean a real or complex number c such that f(c) = 0 . 4. 1. 1 REMARKS. A sharp distinction must be made between the equation f (x) = 0 in the unknown x and the equality f (x) = 0 of polynomials in the indeterminate x. In the former case, x is a definite (though unknown) number and the expression f(x) is also a definite number. Therefore the equation f(x) = 0 is to be correctly interpreted as the condition on this number x that the associated number f(x) should be zero. In the latter case f (x) is a polynomial and so is OJ the equality of these two polynomials means that all coefficients ai of f (x) are zero. Therefore the equality f(x) = 0 of polynomials is the condition on the coefficients ~ of f(x) that they should all be zero. There are many other kinds of equations besides polynomial equations in one unknown. In the first place there are polynomial equations in two or more unknowns x, y, . ... Then there are equations which are not polynomial equations. For example, if f(x) and g(x) =F 0 are polynomials in the indeterminate x, then the equation 66 [35.175.236.44] Project MUSE (2024-03-28 19:02 GMT) Linear, Quadratic and Cubic Equations !(x)/g(x) = 0 would be a rational equation in the unknown X; moreover , unless g(x) is a factor of !(x), it is not a polynomial equation. An expression such as cos2 x+3 sin x+5 = 0 would be a trigonometrical equation in the unknown X, and x+7z +8 =0 would be an exponential equation in the unknown x. Here we are only interested in polynomial equations with real coefficients in one unknown, their properties and their solutions. In the subsequent sections of this chapter we shall use the results of the previous chapters to study the problem (A). To conclude the present section, we observe that...