College algebra . is the product of real linear factors,but if f(x) = 0 has imaginary or complex roots, f(x) containsquadratic factors of the type (x — a)^ + 6 which cannot be sepa-rated into real linear factors. In Art. 95 the graph of /(as) is discussed when the polynomialis the product of real linear factors, and it is shown that, corre-sponding to each linear factor x — r, the graph meets the X-axis at x = r. It should nowbe noted that (a; - ay +b^>0,for all real values of x,and there is, therefore,corresponding to quad-ratic factors of f(x), nointersection of the graphwith the X-axis.
College algebra . is the product of real linear factors,but if f(x) = 0 has imaginary or complex roots, f(x) containsquadratic factors of the type (x — a)^ + 6 which cannot be sepa-rated into real linear factors. In Art. 95 the graph of /(as) is discussed when the polynomialis the product of real linear factors, and it is shown that, corre-sponding to each linear factor x — r, the graph meets the X-axis at x = r. It should nowbe noted that (a; - ay +b^>0,for all real values of x,and there is, therefore,corresponding to quad-ratic factors of f(x), nointersection of the graphwith the X-axis. Example : Graph/ (a) = x - 7 x3 - 4 a;2-I-78 a;= x(x+ 3)(a:2-10x-|-26)= x(x-|-3)[(x-5)2H-l]. Corresponding to the linearfactors X and x + 3, the graphintersects the X-axis at x = 0and X =— 3 respectively (Fig. 29). Corresponding to the quadratic f9,ctorx2—10x4-26 there is no intersection with the X-axis. (In Fig. 29 onehorizontal space represents one unit, while one vertical space representstwenty units.). ^X Fig. 29. Arts. 96-98] TRANSFORMATIONS OF EQUATIONS 131 EXERCISES 1. If ri, rj, ?•?, r„ are roots of an equation, show that (X— J-i)(a-r2)(x-r3)... (x - r„)=0is the equation or an eqtiivalent equation. 2. Poim equations which have the folio-wing roots and no others,(a) 2, 3, 5. (6) 1+2i, 1-2t, wheni2=-l. (c) 1+ V2, 1 - A 3. (d) V2, -V2, V3, -Vs. (e) 1, - 2, 3, 0. (/) 2 + VS, 2 --v/S, - 2 + VF, - 2 -Vs. 3. By means of the theorem concerning the number of roots off(x) = 0,show : (1) if / (x) = 0 he multiplied by a polynomial in Xj the resulting equa-tion has more roots than f(x) = 0; (2) if / (i) = 0 be divided by a polyno-mial in X, which is a factor of f(_x), the resulting equation wiU have fewerroots than / (x) = 0. 4. Plot the graphs of the following : (a)/(x) = (x-l)2(x-3)2.(6)/(x)=x(x-l)(x-4).(c) fix) = (x-l)(x + 2) (X -1- 7).(d)/(x) = (x-f-5)(x-6)2.(e)/(x) = (x-2)2(x-i-2). 5. Show that 3 and ^ are double roots of 9x5 — 51x^ + 58x3-}- 58x2 —51X-1-9 =
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