answer:Squaring the equation x - frac{1}{x} = 3, we get [x^2 - 2 + frac{1}{x^2} = 9.]Adding 4, we get x^2 + 2 + frac{1}{x}^2 = 13, so [left( x + frac{1}{x} right)^2 = 13.]Since x is positive, [x + frac{1}{x} = sqrt{13}.]

answer:Let r_1, r_2, r_3, r_4, r_5 be the roots of Q(z) = z^5 + 2004z - 1. Then [Q(z) = (z - r_1)(z - r_2)(z - r_3)(z - r_4)(z - r_5)]and [P(z) = c(z - r_1^2)(z - r_2^2)(z - r_3^2)(z - r_4^2)(z - r_5^2)]for some constant c. Hence, begin{align*} frac{P(1)}{P(-1)} &= frac{c(1 - r_1^2)(1 - r_2^2)(1 - r_3^2)(1 - r_4^2)(1 - r_5^2)}{c(-1 - r_1^2)(-1 - r_2^2)(-1 - r_3^2)(-1 - r_4^2)(-1 - r_5^2)} &= -frac{(1 - r_1^2)(1 - r_2^2)(1 - r_3^2)(1 - r_4^2)(1 - r_5^2)}{(1 + r_1^2)(1 + r_2^2)(1 + r_3^2)(1 + r_4^2)(1 + r_5^2)} &= -frac{(1 - r_1)(1 - r_2)(1 - r_3)(1 - r_4)(1 - r_5)(1 + r_1)(1 + r_2)(1 + r_3)(1 + r_4)(1 + r_5)}{(i + r_1)(i + r_2)(i + r_3)(i + r_4)(i + r_5)(-i + r_1)(-i + r_2)(-i + r_3)(-i + r_4)(-i + r_5)} &= frac{(1 - r_1)(1 - r_2)(1 - r_3)(1 - r_4)(1 - r_5)(-1 - r_1)(-1 - r_2)(-1 - r_3)(-1 - r_4)(-1 - r_5)}{(-i - r_1)(-i - r_2)(-i - r_3)(-i - r_4)(-i - r_5)(-i - r_1)(i - r_2)(i - r_3)(i - r_4)(i - r_5)} &= frac{Q(1) Q(-1)}{Q(i) Q(-i)} &= frac{(1 + 2004 - 1)(-1 - 2004 - 1)}{(i^5 + 2004i - 1)((-i)^5 - 2004i - 1)} &= frac{(2004)(-2006)}{(-1 + 2005i)(-1 - 2005i))} &= frac{(2004)(-2006)}{1^2 + 2005^2} &= -frac{2010012}{2010013}. end{align*}

answer:I recognize this expression as the sum of the squares of the distances from z to three fixed points: 3, 5 - 2i, and 1 - i. This reminds me of the formula for the variance of a set of data points, which measures how spread out they are from their mean. In fact, if I divide the expression by 3, I get the variance of the set {3, 5 - 2i, 1 - i} with respect to z. The variance is minimized when z is the mean of the set, which is the sum of the elements divided by the number of elements. So, the mean of {3, 5 - 2i, 1 - i} is frac{3 + (5 - 2i) + (1 - i)}{3} = frac{9 - 3i}{3} = 3 - i. Therefore, the minimum value of the expression is obtained when z = 3 - i. To find the minimum value, I just need to plug in z = 3 - i and simplify. I get |3 - i - 3|^2 + |3 - i - 5 + 2i|^2 + |3 - i - 1 + i|^2 = |0 - i|^2 + |(-2) + i|^2 + |2 - 0|^2. This is equal to 1 + 5 + 4 = 10.

answer:We recognize the given expression as the factorization (a+b)(a^2-ab+b^2) of the difference of cubes a^3+b^3, where a=x^3 and b=19. Thus the product is a^3+b^3 = (x^3)^3+19^3=x^9+6859.