Question

What is the quotient, click to see picture

Answer 1

The correct quotient is:
`\[ 2x + 1 + (2x - 5)/(x^2 + x + 1) \]` . Option b is the correct answer.

Polynomial Long Division for 2x^(3)+3x^(2)+5x / x^(2)+x+1:

Step 1: Dividing the leading terms:

First, divide the highest power term of the dividend (2x^3) by the highest power term of the divisor (x^2).

2x | x^2 + x + 1

-----|----------------

2x^3 | 2x^3 + 2x^2 + 2x

| -----------------

2x^2 + 3x

Step 2: Bring down the next term:

Bring down the next term of the dividend (3x^2) next to the remaining term.

2x | x^2 + x + 1

-----|----------------

2x^3 | 2x^3 + 2x^2 + 2x

| -----------------

3x^2 + 5x

Step 3: Divide and multiply:

Divide the remaining term (3x^2) by the leading term of the divisor (x^2) and write the result above the line. Multiply the result by the divisor and place it below the line.

2x + 1 | x^2 + x + 1

-----|----------------

2x^3 | 2x^3 + 2x^2 + 2x

| -----------------

3x^2 + 5x

| -(3x^2 + 3x + 3)

| ------------------

2x + 2

Step 4: Repeat the process:

Repeat the steps of dividing, multiplying, and subtracting until there are no more terms to bring down or the remainder is a polynomial of a lower degree than the divisor.

2x + 1 | x^2 + x + 1

-----|----------------

2x^3 | 2x^3 + 2x^2 + 2x

| -----------------

3x^2 + 5x

| -(3x^2 + 3x + 3)

| ------------------

2x + 2

| -(2x^2 + 2x + 2)

| -------------------

2x - 5

Therefore, the quotient is 2x + 1 and the remainder is 2x - 5. The correct quotient is:
`\[ 2x + 1 + (2x - 5)/(x^2 + x + 1) \]` . Option b is the correct answer.