Question

Using long division method, find the remainder when the polynomial p(x) x³-2x²-x+2 divided by x+1​

Answer 1

Step-by-step explanation:

$\tt\red{ QUESTION}$$\tt\blue{::}$

Using long division method, find the remainder when the polynomial p(x) x³-2x²-x+2 divided by x+1

$\tt\red{ ANSWER}$$\tt\blue{::}$

Polynomials::

Find remainder when x³-2x³+x-2 is divided by (x-2)

Solution:-

Here, f(x)= x³-2x³+x-2 is divided by the linear polynomial (x-2)

At first we will find out the zero of the polynomial (x-2)

=> x - 2 = 0

=> x = 2

From the Remainder Theoreom, we know that when f(x)= x³-2x³+x-2 is divided by (x-2) gives the remainder f(2).

Therefore, The required remainder= f(2)

=> x³-2x³+x-2

=> 2³ - 2. (2)³ + 2 - 2

=> 8 - 2. 8 + 2 - 2

=> 8 - 16 + 2 - 2

=> -8

Hence, The remainder is -8

Answer 2

$\large\underline{\sf{Solution-}}$

Given that

$\purple{\rm :\longmapsto\:Dividend = {x}^{3} - {2x}^{2} - x + 2}$

and

$\purple{\rm :\longmapsto\:Divisor = x + 1}$

So, By using Long Division Method, we have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: {x}^{2} - 3x + 2\:\:}}}\\ {\underline{\sf{x + 1}}}& {\sf{\: {x}^{3}  -  {2x}^{2} - x + 2 \:\:}} \\{\sf{}}& \underline{\sf{- {x}^{3} - {x}^{2} \: \: \:  \:  \:  \:  \:  \:  \:  \: \:\:}} \\ {{\sf{}}}& {\sf{\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  - 3{x}^{2} - x +2  \:  \:  \:  \:   \:  \:  \:  \:\:}} \\{\sf{}}& \underline{\sf{\:\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  3{x}^{2} + 3x  \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: 2x + 2 \:\:}} \\{\sf{}}& \underline{\sf{\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \: - 2x - 2\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: 0\:\:}}  \end{array}\end{gathered}\end{gathered}\end{gathered}$

So,

$\bf\implies \:Remainder = 0$

Verification

$\purple{\rm :\longmapsto\:Dividend = {x}^{3} - {2x}^{2} - x + 2}$

$\purple{\rm :\longmapsto\:Divisor = x + 1}$

$\purple{\rm :\longmapsto\:Remainder = 0}$

$\purple{\rm :\longmapsto\:Quotient = {x}^{2} - 3x + 2}$

Now, Consider

$\rm :\longmapsto\:Divisor \times Quotient + Remainder$

$\rm \:  =  \: (x + 1)( {x}^{2} - 3x + 2) + 0$

$\rm \:  =  \: {x}^{3} - {3x}^{2} + 2x + {x}^{2} - 3x + 2$

$\rm \:  =  \: {x}^{3} - {2x}^{2} - x + 2$

$\rm \:  =  \: Dividend$

Hence, Verified