The Art and Craft of Problem Solving
164 CHAPTER 5 ALGEBRA 5.3.24 Can you generalize the idea used in Exam ple 5.3.2 on page 159? (^1995 1) VIri· Find � 1 f(k)· 5.3 ...
5.4 POLYNOMIALS 165 For example, let f(x) = x^3 +x^2 + 7 and g(x) = x^2 + 3. Both polynomials are in Z[x]. By doing "long divisi ...
166 CHAPTER 5 ALGEBRA The Remainder Theorem If the polynomial P(x) is divided by x - a the remainder will be P(a). For example, ...
5.4 POLYNOMIALS^167 has eight zeros, but only five distinct zeros. The zero 7 appears with multiplicity 3 and the zero - 6 has m ...
168 CHAPTER 5 ALGEBRA Relationship Between Zeros and Coefficients If we multiply out the right-hand side of equation (2) on page ...
5.4 POLYNOMIALS 169 Using the same reasoning, the right-hand side will have 16 monomial terms (before collecting like terms), ea ...
170 CHAPTER 5 ALGEBRA Solution: Let the zeros be a, b, e, d. Then the relationship between zeros and coefficients yields a+b+e+d ...
5.4 POLYNOMIALS 171 If a polynomial with integer coefficients can be fa ctored into polyno mials with rational coefficients, it ...
172 CHAPTER 5 ALGEBRA For the first step, observe that if n > 1, then n < J n2 + 1 < n + 1/ n. The first inequality is ...
5.4.9 Find the remainder when you divide x^81 +x^49 + x^25 +x^9 +x by x^3 -x. 5.4.10 Let p(x) = x^6 + x^5 + ... + 1. Find the re ...
174 CHAPTER 5 ALGEBRA SO b^2 + b + 1 lies strictly between two consecutive perfect squares. An impossibility! These examples use ...
5.5 INEQUALITIES 175 In summary, the hierarchy of growth rates, from slowest to highest, is logarithms, powers, exponents. Simpl ...
176 CHAPTER 5 ALGEBRA Now our problem is equivalent to determining the relative order of f( 1998) and f( 1999). How does this fu ...
5.5 INEQUALITIES 177 Consider the following equivalent formulation. Let x,y be positive real numbers with sum S = x + y. Then th ...
178 CHAPTER 5 ALGEBRA As the distance between two positive numbers decreases, their product increases, provided that their sum s ...
5.5 INEQUALITIES 179 begin by restating AM-GM with its equivalent "sum and product" formulation.^15 5.5. 17 AM-GM Reformulated. ...
180 CHAPTER 5 ALGEBRA The AM-GM inequality is the starting point for many interesting inequalities. Here is one example (see the ...
5.5 INEQUALITIES 181 Thus the n^2 terms can be decomposed into n terms that equal 1 and (n^2 - n) /2 pairs of terms, with each p ...
182 CHAPTER 5 ALGEBRA We conclude that (^10000 1) 2 VlOOO I - 2 < n�^1 vn <^2 VlOOOO. This tells us that l A J is either 1 ...
Thus, for any real a, b, c, x, y, z, we have 5.5 INEQUALITIES 183 O:S (ay - bxf + ( az -cx)^2 + (bz -cy)^2. This is equivalent t ...
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