Question

Apply the division algorithm to find the quotient and the remainder on dividing f(X) by g(x) as given below:
$f(x) = {x^3} - 6{x^2} + 11x - 6,{\text{ g(x) = x + 2}}$

Answer 1

Hint – In this question put${x^3} - 6{x^2} + 11x - 6 = \left( {x + 2} \right)\left( {a{x^2} + bx + c} \right) + R$, using the division algorithm, then simplify the R.H.S part and proceed by comparing the coefficients both the sides.

Complete step-by-step answer:

As we know that the division algorithm states that:

Dividend = divisor $ \times $Quotient + Remainder

Now it is given that the dividend is ${x^3} - 6{x^2} + 11x - 6$ and the divisor is (x + 2).

So let the quotient be $q\left( x \right) = a{x^2} + bx + c$ and the remainder $r\left( x \right)

 = R$. Therefore using division algorithm we have,

$ \Rightarrow {x^3} - 6{x^2} + 11x - 6 = \left( {x + 2} \right)\left( {a{x^2} + bx + c} \right) + R$

Now simplify the R.H.S part we have,

$ \Rightarrow {x^3} - 6{x^2} + 11x - 6 = \left( {a{x^2} + bx + c} \right)x + \left( {a{x^2} + bx + c}

\right)2 + R$

$ \Rightarrow {x^3} - 6{x^2} + 11x - 6 = \left( {a{x^3} + b{x^2} + cx} \right) + \left( {2a{x^2} +

2bx + 2c} \right) + R$

$ \Rightarrow {x^3} - 6{x^2} + 11x - 6 = a{x^3} + \left( {b + 2a} \right){x^2} + \left( {c + 2b}

\right)x + 2c + R$

Now comparing the coefficient we have,

$ \Rightarrow a = 1$........... (1)

$\left( {b + 2a} \right) = - 6$...................... (2)

$\left( {c + 2b} \right) = 11$.................. (3)

$\left( {2c + R} \right) = - 6$................... (4)

Now substitute the value of a from equation (1) in equation (2) we have,

 $ \Rightarrow \left( {b + 2} \right) = - 6$

$ \Rightarrow b = - 6 - 2 = - 8$............................ (5)

Now substitute the value of b from equation (5) in equation (3) we have,

$ \Rightarrow \left( {c + 2\left( { - 8} \right)} \right) = 11$

$ \Rightarrow c = 11 + 16 = 27$.............................. (6)

Now substitute the value of c from equation (6) in equation (4) we have,

$ \Rightarrow \left( {2\left( {27} \right) + R} \right) = - 6$

$ \Rightarrow R = - 6 - 54 = - 60$

So the quotient becomes

$ \Rightarrow q\left( x \right) = a{x^2} + bx + c = {x^2} - 8x + 27$

And the remainder becomes

$r\left( x \right) = R = - 60$

So this is the required answer.

Hence option (C) is correct.


Note – The division algorithm is an algorithm that takes two integers N and D and helps in computation of their quotient and remainder part, $\left( {N/D} \right) = \left( {Q,R} \right)$. The key point is the coefficient comparison is only done when both sides are equal and the comparison is done upon the terms with same powers on both sides.