Question

Look at the work shown for the division problem shown on the right. The remainder is . Divide 2x4- 4x3 - 11x2 + 3x - 6 by x + 2.

Answer 1

To divide 2x^4 - 4x^3 - 11x^2 + 3x - 6 by x + 2, you can use long division.

1. Write the division problem in long division format, with the dividend (2x^4 - 4x^3 - 11x^2 + 3x - 6) inside the division symbol and the divisor (x + 2) outside the division symbol.

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x + 2 | 2x^4 - 4x^3 - 11x^2 + 3x - 6

2. Divide the first term of the dividend by the first term of the divisor. In this case, divide 2x^4 by x to get 2x^3. Write this result above the division symbol.

2x^3

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x + 2 | 2x^4 - 4x^3 - 11x^2 + 3x - 6

3. Multiply the divisor (x + 2) by the result (2x^3). In this case, multiply (x + 2) by (2x^3) to get 2x^4 + 4x^3. Write this result below the dividend, aligned with the like terms.

2x^3

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x + 2 | 2x^4 - 4x^3 - 11x^2 + 3x - 6

- (2x^4 + 4x^3)

4. Subtract the result obtained in step 3 from the dividend. In this case, subtract (2x^4 + 4x^3) from (2x^4 - 4x^3 - 11x^2 + 3x - 6) to get -8x^3 - 11x^2 + 3x - 6. Write this result below the line.

2x^3

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x + 2 | 2x^4 - 4x^3 - 11x^2 + 3x - 6

- (2x^4 + 4x^3)

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- 8x^3 - 11x^2 + 3x - 6

5. Bring down the next term from the dividend, which is -8x^3. Write this term next to the result obtained in step 4.

2x^3 - 8x^3

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x + 2 | 2x^4 - 4x^3 - 11x^2 + 3x - 6

- (2x^4 + 4x^3)

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- 8x^3 - 11x^2 + 3x - 6

6. Repeat steps 3 and 4 with the new dividend (-8x^3 - 11x^2 + 3x - 6).

7. Continue the process of division until you have brought down all the terms of the dividend and have no remainder left.

The final result will be the quotient obtained from the division, which in this case will be 2x^3 - 8x - 5, and the remainder will be -16.

Answer 2

The quotient is 2x³ - 8x²+ 5x - 16 with a remainder of 26.The remainder of 26 indicates that when the polynomial 2x⁴ - 4x³ - 11x² + 3x - 6 is divided by x + 2, the resulting expression is not completely divisible. The quotient 2x^3 - 8x^2 + 5x - 16 represents the part that is evenly divisible, while the remainder 26 is what's left after the division.

Step-by-step explanation:

To divide the polynomial 2x⁴ - 4x³ - 11x² + 3x - 6 by x + 2, we perform polynomial long division. Start by dividing the leading term of the numerator (2x^4) by the leading term of the denominator (x). The result is 2x^3. Multiply the entire divisor (x + 2) by this quotient and subtract it from the original polynomial. Repeat this process for each term until the remainder is of a lower degree than the divisor.

In this case, the division leads to the quotient 2x³ - 8x²+ 5x - 16 and a remainder of 26. Therefore, the final answer is (2x³ - 8x² + 5x - 16) with a remainder of 26 when divided by (x + 2).

The polynomial division can be expressed as follows:

`\[ (2x^4 - 4x^3 - 11x^2 + 3x - 6)/(x + 2) = 2x^3 - 8x^2 + 5x - 16 + (26)/(x + 2) \]`

This process ensures that the remainder is of a lower degree than the divisor, and the quotient represents the result of the division.