Question

Find the polynomial of least degree which should be subtracted from the polynomial x4 + 2x3 – 4x2 + 6x – 3 so that it is exactly divisible by x2 – x + 1.

Answer 1

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Answer 2

The required polynomial of least degree is x - 1

Given :

The polynomial x⁴ + 2x³ - 4x² + 6x - 3

To find :

The polynomial of least degree which should be subtracted from the polynomial x⁴ + 2x³ - 4x² + 6x - 3 so that it is exactly divisible by x² – x + 1.

Solution :

Step 1 of 3 :

Write down the given polynomial

Here the given polynomial is x⁴ + 2x³ - 4x² + 6x - 3

Step 2 of 3 :

Perform the division

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: {x}^{2} + 3x - 2 \:\:}}}\\ {\underline{\sf{ {x}^{2} - x + 1 }}}& {\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: {x}^{4} + 2 {x}^{3} - 4 {x}^{2} + 6x - 3 }} \\{\sf{}}& \underline{\sf{ \: \: \: {x}^{4} - {x}^{3} + {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: }} \\ {\underline{\sf{}}}& {\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 3 {x}^{3} - 5 {x}^{2} + 6x \: \: \: \: \: \: \: \: \: }} \\{\sf{}}&\underline{\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 3 {x}^{3} - 3 {x}^{2} + 3x \: \: \: \: \: \: }} \\ {\underline{\sf{}}}&{\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 2 {x}^{2} + 3x - 3 \:\:}} \\{\sf{}}&\underline{\sf{\:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 2 {x}^{2} + 2x - 2\:\:}} \\ {\underline{\sf{}}}& {\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: x - 1}} \\{\sf{}}\\ {\sf{\: \: \: \: \: \: \: \: \: \: \:\:}}{\sf{}}&{\sf{\:\:\:\:}}\end{array}\end{gathered}$

Step 3 of 3 :

Find the polynomial

We see that when we divide x⁴ + 2x³ - 4x² + 6x - 3 by the polynomial x² – x + 1 the remainder is x - 1

Hence the required polynomial of least degree is x - 1

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