Roots

Example: If the roots of x^3 + bx^2 + cx + d = 0 are -1, 2, 5, then b + c + d equals:
One method could be to expand (x+1)(x-2)(x-5) and then add b,c and d but there is a simpler method of obtaining the answer. Let's start with b.

-b/a is the sum of the roots. since a is 1, -b= -1 + 2 + 5. b=-6

c/a is the result of adding the product of the roots taken two at a time. (-1)(2) + (-1)(5) + (2)(5) = c/a.
c =3

-d/a is the product of the roots. -d/a = (-1)(2)(5). d = 10

b + c + d = -6 + 3 + 10 = 7

-Kevin

Changing from Base 10 to Base 2

(This is Marisa!)
So I JUST discovered how to convert numbers from Base 10 to Base 2, which will occasionally show up on both the Mathematics and the Number Sense Tests! The easiest way to show you how to convert numbers is by solving a problem myself, step by step.


Let's say you would like to convert 46210 to ___2. First, find the largest multiple of 2 that can divide into 462. After that, begin your division!


1. 462/256 (Largest multiple of 2 that is less than 462) = 1, with a remainder, R, of 206. For each following step, you will take your remainder and divide it by the next smallest multiple of 2, until you reach zero.


2. 206/128 = 1R78.


3. 78/64 = 1R14.


4. 14 Cannot be divided by 32 or 16, but CAN be divided by 8. Therefore, remember the first two zeroes and divide by 8, continuing as normal. 14/8 = 1R6.


5. 6/4 = 1R2.


6. 2/2 = 1R0.


7. 0/1 = 0.


Once you've reached 0, go back and write down all of your answers, ignoring the remainders. Remember the two zeroes in step 4! You should end up with 111001110.


There you have it! To sum it up, 46210 is equal to 1110011102.