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However, according to the remainder theorem, the remainder is the value of the cubic function when you substitute x = 2.
(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)
x3-x2
x3+3x2+6x+1 divided by x+1 Quotient: x2+2x+4 Remaider: -3
x3 - x2 + 2x = x*(x2 - x + 2) which cannot be factored further.
x3 - x2 + x - 2 has no rational factors.
1.6667
Dividend: 4x4-x3+17x2+11x+4 Divisor: 4x+3 Quotient: x3-x2+5x-1 Remainder: 7
(x3 + 4x2 - 3x - 12)/(x2 - 3) = x + 4(multiply x2 - 3 by x, and subtract the product from the dividend)1. x(x2 - 3) = x3 - 3x = x3 + 0x2 - 3x2. (x3 + 4x2 - 3x - 12) - (x3 + 0x2 - 3x) = x3 + 4x2 - 3x - 12 - x3 + 3x = 4x2 - 12(multiply x2 - 3 by 4, and subtract the product from 4x2 - 12)1. 4x(x - 3) = 4x2 - 12 = 4x2 - 122. (4x2 - 12) - (4x2 - 12) = 4x2 - 12 - 4x2 + 12 = 0(remainder)
If x2 and x3 are meant to represent x2 and x3, then x2 times x3 = x5 You find the product of exponent variables by adding the exponents.
x3 + 9x2 + 27x + 27 Given the numbers in the equation, we can likely bet on (x + 3) being a factor. Let's try it with artificial division: 3 * 1 = 3 9 - 3 = 6 3 * 6 = 18 27 - 18 = 9 3 * 9 = 27 27 - 27 = 0 Bingo. So let's carry it out in long division:                       x2 + 6x + 9                    _____________________ x + 3 ) x3 + 9x2 + 27x + 27                        x3 + 3x2                                        6x2 + 27x                                        6x2 + 18x                                                                9x + 27                                                                9x + 27                                                                                    0 So we have: x3 + 9x2 + 27x + 27 = (x + 3)(x2 + 6x + 9) Which we can now factor further with relative ease: = (x + 3)(x + 3)(x + 3) = (x + 3)3
(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)
x3-x2
(-x3 + 75x - 250) / (x + 10) = x2 - 10x - 25
x3+3x2+6x+1 divided by x+1 Quotient: x2+2x+4 Remaider: -3
x3 /12 + 16x + c
X3 X(X2) X2(X) and, X * X * X
Dividend: 4x^4 -x^2 +17x^2 +11x +4 Divisor: 4x +3 Quotient: x^3 -x^2 +5x -1 Remainder: 7