Advanced High-School Mathematics

SECTION 3.6 Cubic Discriminant 171 ∆<0 =⇒P(x) has one real zero and two non-real complex zeros. This is all rounded out by th ...

172 CHAPTER 3 Inequalities Example. Compute the minimum value of the function f(x) = 1 x^2 +x, x >0. Solution. The minimum va ...

SECTION 3.6 Cubic Discriminant 173 1 2 3 2 4 6 x y x^3 −xy+2=0 y+x =c 2 level curves Equation 1: y=2/x+x² Equation 2: y=c−x² Equ ...

174 CHAPTER 3 Inequalities R(f) = det       a b c d 0 0 a b c d 3 a 2 b c 0 0 0 3a 2 b c 0 0 0 3a 2 b c      ...

SECTION 3.7 Discriminant 175 V = det       1 x 1 x^21 ··· xnn−^1 1 x 2 x^22 ··· xn 2 −^1 ... ... 1 xn x^2 n ··· xnn−^ ...

176 CHAPTER 3 Inequalities From the above, we see that ∆ = ∆(f) =a^2 nn−^2 ∏ 1 ≤i<j≤n (xj−xi)^2. The difficulty with the abov ...

SECTION 3.7 Discriminant 177 S(f,g) =               an an− 1 an− 2 ··· 0 0 0 0 an an− 1 ··· 0 0 0 ... ...

178 CHAPTER 3 Inequalities Next, write f(x) =an(xn+a′n− 1 xn−^1 +···+a′ 1 x+a′ 0 ), a′i=ai/an, i= 0, 1 ,...,n−1; similarly, writ ...

SECTION 3.7 Discriminant 179 R(g,h) = det         1 an− 1 ··· a 1 a 0 0 ··· 0 0 0 ... S(g,h) ... 0      ...

180 CHAPTER 3 Inequalities zerosα 1 , ..., αn. Then R(f,g) =amn ∏n i=1 g(αi). Proof. Letg(x) have zerosβ 1 ,...,βm. Since g(x) = ...

SECTION 3.7 Discriminant 181 where the convention is that the factor under the” is omitted. From the above, we see immediately t ...

182 CHAPTER 3 Inequalities 3.7.3 A special class of trinomials We shall start this discussion with a specific example. Let f(x) ...

SECTION 3.7 Discriminant 183 it has the coefficients of g(x). Note that adding a multiple a of row m+n to the firstmrows ofS(f,g ...

184 CHAPTER 3 Inequalities y=3/x+x y=1/x 3 y=x 3 y=m We let m be the minimum value of f(x) and note that the graph of y=mmust be ...

Chapter 4 Abstract Algebra While an oversimplification,abstract algebragrew out of an attempt to solve and otherwise understand ...

186 CHAPTER 4 Abstract Algebra Operations on subsets of a given set: intersection (∩), union, (∪),difference(−), andsymmetric d ...

SECTION 4.1 Basics of Set Theory 187 we see thatR∈Rif any only ifR6∈R! This is impossible! This is a paradox, often called Russe ...

188 CHAPTER 4 Abstract Algebra Z⊆Q⊆R⊆C. As a more geometrical sort of example, let us consider the setR^3 of all points in Carte ...

SECTION 4.1 Basics of Set Theory 189 Number of subsets = number of subsets of size 0 + number of subsets of size 1 + number of s ...

190 CHAPTER 4 Abstract Algebra Determine which of the following real numbers are inQ(2): π, 2 3 , 10 2 , cos(π/4), 12 , 3 4 , 12 ...