Question

Divide the polynomial 3x4 – 4x3 – 3x –1 by (x-1) and find the quotient and remainder. Also, verify the remainder by applying Remainder Theorem. Please help its urgent..

Answer 1

Here is ur answer

Hope it helps

Verification included

Answer 2

The quotient of division is $\bf 3 {x}^{3} - {x}^{2} - x - 4$ and remainder is -5.

It has been verified by remainder theorem too.

Given:

• $3 {x}^{4} - 4 {x}^{3} - 3x - 1$
• $x - 1$

To find:

• Perform division, find the quotient and remainder.
• Also, verify the remainder by applying Remainder Theorem.

Solution:

Theorem to be used:

Remainder Theorem: If polynomial p(x) is divided by x-a, then remainder is given by p(a).

Step 1:

Perform long division.

$x - 1 \: ) \: 3 {x}^{4} - 4 {x}^{3} - 3x - 1 \: (3 {x}^{3} - {x}^{2} - x - 4 \\ 3 {x}^{4} - 3 {x}^{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \\ ( - ) \: \: \: ( + ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - \: \: \: \: \: \: \: \: \: \\ - {x}^{3} \: \: \: \: \: - 3x \\ - {x}^{3} + {x}^{2} \: \: \: \: \: \: \\ ( + ) \: \: \: ( - ) \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - \: \: \: \: \: \: \\ - {x}^{2} - 3x \\ - {x}^{2} + x \\ ( + )( - ) \\ - - - - \\ - 4x - 1 \\ - 4x + 4 \\ ( + )( - ) \\ - - - - - \\ - 5 \\ - - - - - -$

Thus,

Quotient is $\bf 3 {x}^{3} - {x}^{2} - x - 4$ and remainder is -5.

Step 2:

Apply remainder theorem.

Let

$p(x) = 3 {x}^{4} - 4 {x}^{3} - 3x - 1$

Put x= 1 in the polynomial.

$p(1) = 3 {(1)}^{4} - 4 {(1)}^{3} - 3(1) - 1$

or

$p(1) = 3 - 4 - 3 - 1$

or

$\bf p(1) = - 5$

It has been verified by remainder theorem too.

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