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Chapter 2: Problem 15

Use long division to divide. $$\left(x^{4}+5 x^{3}+6 x^{2}-x-2\right) \div(x+2)$$

Short Answer

Expert verified
The result of the division is \(x^{3}+3x^{2}-x+1\).

Step by step solution

01

Set Up the Long Division

First, set up the long division with the dividend \(x^{4}+5 x^{3}+6 x^{2}-x-2\) on the inside of the equation, and the divisor \(x+2\) on the outside. Ensure the terms of the polynomial are in decreasing order of power.
02

Divide the First Term

Divide the first term of the dividend \(x^{4}\) by the first term of the divisor \(x\). The result \(x^{3}\) is the first term of the quotient.
03

Multiply and Subtract

Next, multiply the divisor \(x+2\) by the first term of the quotient \(x^{3}\) we obtained in step 2. Subtract this result from the dividend \(x^{4}+5 x^{3}+6 x^{2}-x-2\) and bring down the next term from the original dividend. This subtraction will give a new dividend.
04

Repeat the Process

Repeat step 2 for the new dividend, constantly updating the quotient and the dividend. This process of dividing, multiplying, and subtracting continues till all the terms in the original dividend have been brought down.
05

Write the Final Result

Once completed the division, what remains of the dividend is the remainder, and the quotient contains the result of the division. Write down the final result which includes the quotient and the remainder, if present.

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