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Q: When dividing 2x2 - 4x plus 5 by x - 1 the remainder is 11?

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The answer depends on what you are dividing by 11.

-11

There is no rational factorisation to this expression.

(x3 + 3x2 - 2x + 7)/(x + 1) = x2 + 2x - 4 + 11/(x + 1)(multiply x + 1 by x2, and subtract the product from the dividend)1. x2(x + 1) = x3 + x22. (x3 + 3x2 - 2x + 7) - (x3 + x2) = x3 + 3x2 - 2x + 7 - x3 - x2 = 2x2 - 2x + 7(multiply x + 1 by 2x, and subtract the product from 2x2 - 2x + 7)1. 2x(x + 1) = 2x2 + 2x2. (2x2 - 2x + 7) - (2x2 + 2x) = 2x2 - 2x + 7 - 2x2 - 2x = -4x + 7(multiply x + 1 by -4, and subtract the product from -4x + 7)1. -4(x + 1) = -4x - 42. -4x + 7 - (-4x - 4) = -4x + 7 + 4x + 4 = 11(remainder)

x = (-1 ± √11) / 2

There are 11 possible remainders (1 to 11). If something is divisible by 12, there is said to be no remainder, but this can be considered to be a remainder of 0, making 12 possible remainders.

If the remainder must be an integer value, then the largest remainder you can have from any equation when dividing by 12 is a remainder of 11. Example: (23 / 12 = 1 R11) 12 goes into 23 one time, and there's 11 left over. If there was more than 11, such as (24 / 12), then the answer would be 2 with no remainder.

6a + 5a = -11 Combining like terms: 11a = -11 Dividing by 11: a = -1

The roots of the equation are [5 +/- sqrt(11)]/2 = 4.158 and 0.842

A subtle point is that the remainder is always a positive number. So -7 divided by 3 is -3, remainder 2 (rather than -2, remainder -1). This is important for the date of Easter because we must often calculate the remainder when dividing into a negative number. Since in that case we are not interested in the quotient it is sufficient to calculate the negative remainder and then simply add to it the number we were dividing by. So -11 mod 3 may be calculated as: -11 divided by 3 is -3, remainder -2, so the number required is -2 + 3 = 1.

It does if you are counting in base 9. 5 plus 5 equals 1 nine and a remainder of 1.

11 / 305: quotient = 0, remainder = 11

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