PurpleWasp

(x ^ 3 + 8x ^ 2 + 21x + 18) can be factored into ( x + 2 )( x + 3 )( x + 3 ). We divide by ( x + 2 ), and they cancel. Expanding

( x + 3 ) ^ 2 gives us x ^ 2 + 6x + 9. A = 1, B = 6, and C = 9. D is the one value that x can't be. If x was -2, then we would be dividing by 0, which absolutely can't happen. So, D = -2. Adding them together, A + B + C + D = 1 + 6 + 9 + (-2) = 14.

The two youngest kids have four pieces each. That leaves 5 pieces to distribute to 5 kids. Each piece of candy can go to any of the 5 kids, so 5^5, or  5*5*5*5*5, or 3125 ways.

That's incorrect. Thanks, tho!

The new area of Square A is (2009 +x)^2, while the new area of Square B is (2009 - x)^2. The difference is \(\begin{align*} &(2009+x)^2-(2009-x)^2\\ &\qquad=(2009+x+2009-x)(2009+x-2009+x) \\ &\qquad=(2\cdot 2009)(2x) \end{align*}\)

For this to be at least as great as the area of a 2009 by 2009 square, \(2(2009)2(x)\geq 2009^2\Rightarrow x\geq \boxed{\frac{2009}{4}}.\)

Thanks so much!

Let a and b] be the roots of this polynomial. Therefore, a+b=13 and ab=4. Notice that a^2+ab+b^2. This means that the sum of the squares of the roots can be obtained by subtracting the term containing the product of a and b, so a^2+b^2=169-2(4)=161.