The question concerns a cubic (order 3) divided by a binomial (order 1) and so the quotient has order 3 - 1 = 2 and the remainder has order 0 (one less than the denominator).
That is
(2*x^3 + 5*x^2 - 14*x + 3)/(2*x - 3) = A*x^2 + B*x + C + D/(2*x - 3)
Multiplying both sides by 2x - 3 gives
2*x^3 + 5*x^2 - 14*x + 3 = 2A*x^3 - 3A*X^2 + 2B*x^2 - 3B*x + 2C*x - 3C + D
Compare coefficients of x^3: 2 = 2A => A = 1
Compare coefficients of x^2: 5 = -3A + 2B = -3 + 2B => 2B = 8 => B = 4
Compare coefficients of x^1: -14 = -3B + 2C = -12 + 2C => 2C = -2 => C = -1
Compare coefficients of x^0: 3 = -3C + 2D = 3 + 2D => 2D = 0 => D = 0
So, The quotient is (x^2 + 4 x - 1) and the remainder is 0.
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Alternatively you can do the long division by considering the coefficients of the largest power of x - ie dividing the largest power of x (left in the dividend) by 2x (the largest power of x in the divisor) and using that as the value to add into the quotient and to multiply the divisor by before subtracting it (as in a normal long division):
_______________ x² + _4x - 1
_______--------------------------
2x - 3 | 2x³ + 5x² - 14x + 3 ___ ← 2x³ ÷ 2x = x²; put "x²" in quotient
________ 2x³ - 3x² ______________ ←(2x - 3) × x² and subtract
________ ------------
______________ 8x² - 14x ________ ← bring down "- 14x"; 8x² ÷ 2x = 4x; put "+ 4x" in quotient
______________ 8x² - 12x ________ ← (2x - 3) × 4x and subtract
______________ ------------
____________________ -2x + 3 _____ ← bring down "+ 3"; -2x ÷ 2x = -1; put "- 1" in quotient
____________________ -2x + 3 _____ ← (2x - 3) × -1 and subtract
_____________________ ---------
__________________________ 0 _____ ← nothing to bring down, result is 0, so no remainder
(The underscores in the diagram are to help align the workings.)
→ (2x³ + 5x² - 14x + 3) ÷ (2x - 3) = x² + 4x - 1
6x3+29x2-40x-42 divided by 6x+5 Quotient: x2+4x-10 Remainder: 8
x3+3x2+6x+1 divided by x+1 Quotient: x2+2x+4 Remaider: -3
Dividend: 6x^3 +29^2 -40x -42 Divisor: 6x +5 Quotient: x^2 +4x -10 Remainder: 8
The quotient of a reciprocal would equal the numerator squared divided by the denominator squared. i.e. (x/y)/(y/x) = x2/y2
Dividend: 6x^3+29x^2-40x-42 Divisor: 6x+5 Quotient: x^2+4x-10 Remainder: 8
6x3+29x2-40x-42 divided by 6x+5 Quotient: x2+4x-10 Remainder: 8
The quotient works out as: x^2+2x+4 and there is a remainder of -3
x3+3x2+6x+1 divided by x+1 Quotient: x2+2x+4 Remaider: -3
Dividend: 6x^3 +29^2 -40x -42 Divisor: 6x +5 Quotient: x^2 +4x -10 Remainder: 8
When any number is divided by itself, the quotient is always ' 1 ' exactly.
The quotient of a reciprocal would equal the numerator squared divided by the denominator squared. i.e. (x/y)/(y/x) = x2/y2
1
Dividend: 6x^3+29x^2-40x-42 Divisor: 6x+5 Quotient: x^2+4x-10 Remainder: 8
Dividend: 4x^4 -x^3 +17x^2 +11x +4 Divisor: 4x +3 Quotient: x^3 -x^2 +5x -1 Remainder: 7
its 5 squared divided by the sum of 28 to the power of ten. -maths teacher
you divided and you are .
2x4 - 9x3 + 13x2 - 15x + 9 = 2x4 - 6x3 - 3x3 + 9x2 + 4x2 - 12x - 3x + 9 = 2x3(x - 3) - 3x2(x - 3) + 4x(x - 3) - 3(x - 3) = (x - 3)*(2x3 - 3x2 + 4x - 3) So the quotient is (2x3 - 3x2 + 4x - 3) and the remainder is 0.