What is the quotient and remainder if any when 2x cubed plus 5x squared -14x plus 3 is divided by 2x -3 showing work?

Answer 1

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