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Chapter 4: Problem 8

In Exercises 5鈥�10, divide using polynomial long division. $$ \left(x^3+x^2+x+2\right) \div\left(x^2-1\right) $$

Short Answer

Expert verified
The result of the division \((x^3 + x^2 + x + 2) \div (x^2 - 1)\) is \(x + x + 3 + \frac{5}{x^2-1}\)

Step by step solution

01

Set up the division

Set up the division in the same way that you might set up a long division problem with numbers. Write \(x^3+x^2+x+2\) inside the division bracket and \(x^2-1\) outside the division bracket.
02

Divide the first term in the dividend by the first term in the divisor

Divide the first term of the polynomial inside the bracket \(x^3\) by the first term of the polynomial outside the bracket \(x^2\). This yields \(x\). Write this above the division bracket.
03

Multiply, subtract and bring down

Now, multiply the divisor \(x^2-1\) by the result from Step 2, which is \(x\), and then subtract this from the current dividend \(x^3+x^2+x+2\). The result of this subtraction replaces the original dividend and brings down the next term from the original dividend to form a new dividend.
04

Repeat the process

Repeat the process started in Step 2 with the new dividend. This will give you the final answer.
05

Write the Remainder

Finally, any term that remains undivided is the remainder. If there's any, then the result of the division is expressed as 'quotient + remainder/divisor'. If there's no remainder, then the quotient is the final answer.

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