College algebra . 27)makes it of much service in the theory of equations. EXERCISES Construct the graphs of the followingfunctions and locate their real zeros approxi-mately (to within ). 1. /(a!) = x3-6a;2-|-lla! pointed out in Art. 89, synthetic divi-sion furnishes a convenient method ofevaluating /(x) for different values of () is obtained as follows :1-6 + 11- 6 + ,/() = this way the following values areobtained: /(-2) = -60. /() = /(-I) = -24. /(2)=0. /() = /() = /(0)^-6. /(3)=0. / (
College algebra . 27)makes it of much service in the theory of equations. EXERCISES Construct the graphs of the followingfunctions and locate their real zeros approxi-mately (to within ). 1. /(a!) = x3-6a;2-|-lla! pointed out in Art. 89, synthetic divi-sion furnishes a convenient method ofevaluating /(x) for different values of () is obtained as follows :1-6 + 11- 6 + ,/() = this way the following values areobtained: /(-2) = -60. /() = /(-I) = -24. /(2)=0. /() = /() = /(0)^-6. /(3)=0. / () = /(4)=6. /(1)=0. /(5) = 24. The graph is shown in Fig. 26; it presentsto the eye the following facts: (1) /(x) has zeros at 1, 2, and 3. (2) fix) is positive when 2 > 36 > 1, andwhenx>3. > (3) / (x) is negative when x x>2. 2. x3 —6x2 + 8x. 3. x3 —7x2 + 14x —8. 4. x-6x + 4. 5. X* —2x3 —7x2+8x +12. 6. x4-3x2 + 6x —6. 7. 3x<-4x3-12x2 + 3. 8. x3 — 126 THEORY OF EQUATIONS [Chap. XIII. 92. General equation of degree n. By equating to zero the gen-eral polynomial of the nth degree, we obtain what is known as thegeneral equation of the nth degree in one unknown. That is to say, OaH + aja;-! + a^x-^ -\ 1- a„ = 0 is the general equation of the nth degree. The principal object of this chapter is to present methodswhich aid in determining exactly or approximately the real rootsof special numerical * equations included under this type. Itis largely for this purpose that we discuss the graphs of poly-nomials. The zeros of the polynomial are the roots of the equa-tion formed by equating the polynomial to zero. The real rootsof the equation may then be looked upon geometrically as theabscissas of the points of the X-axis where the graph of the poly-nomial cuts this axis. 93. Factor theorem. Ifr is a root of the equation f(x) = 0, thenX — r is a factor off(x). Since a zero of f(x) is a root of the equation f(x) = 0, thistheorem follows
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