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1.9. Other Problems 43 If y is nonzero, deduce that Y + (b2 - Wq + q2)/y + 2(ap - q) = o. Find the square roots of 1 - x + d22x ...
44 1. Fundamentals How many distinct terms are there in the expansion of 2&l + x2)(21 + x2 + 23) *. * (Xl + x2 +.. * + zn) ...
Hints 45 1.20. p(t2) has terms in only even powers of t. 2.3. One root is t = m. 2.5. The discriminant should vanish. 2.8. The e ...
46 1. Fundamentals 7.7. A complete collection of quadratic polynomials which can be ex- pressed as the product of two linears ca ...
Hints^47 (i) i = cos(?r/2) + isin(?r/2), so that i(cos 4 + i sin 4) = cos(+ + x/2) + i sin(d + 7r/2); (ii) cos 4 = sin(d + x/2) ...
48 1. Fundamentals 9.12. Let the given polynomial be equal to c[fi(~,y)]~. Since constant terms of the fi must vanish, we can wr ...
2 Evaluation, Division, and Expansion 2.1 Homer’s Method A Knight wishes to evaluate the polynomial 8t3 - 5t2 + 4t + 1 at t = 2. ...
50 2. Evaluation, Division, and Expansion (b) A paper-and-pencil algorithm for making the evaluation in (a) has the following ta ...
2.1. Horner’s Method^51 In applying Horner’s Method, you should not fail to record zero coef- ficients. Check that the table fo ...
52 2. Evaluation, Division, and Expansion (c) Construct the Horner table for evaluating at t = 3 the polyno- mial t3 + 3t2 + lot ...
2.1. Horner’s Method 53 Expand the polynomial y7 - 4y6 + 2y4 - y” + y + 1 in terms of (y + 2) and evaluate it at y = -1.95 to t ...
54 2. Evaluation, Division, and Expansion Taking notice of the fact that the nth term of any series is equal to the difference b ...
2.1. Horner’s Method^55 show that c n ,.(k) _ ’ r=l h+l(n + l)(k+‘) fork= 1,2,3 ,.... Since we know how to sum factorial power ...
56 2. Evaluation, Division, and Expansion For the polynomial 4r3 + 2r2 - r - 1, verify that the Horner’s table is 4 2 -1 -1 0 0 ...
2.2. Division of Polynomials 57 where q(t) is a polynomial and k a constant polynomial over D. (i) Show that degq = (degp) - 1. ...
58 2. Evaluation, Division, and Expansion If we think of the “size” of a polynomial as being measured by its de- gree, then, as ...
2.2. Division of Polynomials 59 8 9 10 11 12 13 Explain how to perform the algorithm, how to read off the quotient, and remainde ...
60 2. Evaluation, Division, and Expansion (a) Set z = 0 and obtain that f (x, y, 0) = (x + y)x29. (b) Consider the polynomial Ll ...
2.2. Division of Polynomials 61 (c) In (b), argue that P(h)..., tn) -q(s1,s2 ,..., h-1) is a symmetric polynomial which vanishes ...
62 2. Evaluation, Division, and Expansion Such an array of vertices and edges is called a graph. Some examples of graphs are the ...
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