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Chapter 2: Problem 21

Use long division to divide. $$\left(x^{3}-9\right) \div\left(x^{2}+1\right)$$

Short Answer

Expert verified
The solution to \((x^{3}-9) \div (x^{2}+1)\) is \(x-\frac{x+9}{x^{2}+1}\)

Step by step solution

01

Set up the divison

Write the problem as division: \[x^{3}-9\div x^{2}+1\]. The cubic polynomial \(x^{3}-9\), which we are dividing by \(x^{2}+1\), is the dividend and \(x^{2}+1\) is the divisor.
02

Divide the first term

Divide the leading term of the dividend, \(x^{3}\), by the leading term of the divisor, \(x^{2}\), to get \(x\). This is the first term of the quotient.
03

Multiply and Subtract

Multiply the divisor \(x^{2}+1\) by the first term of the quotient, \(x\), to get \(x^{3}+x\). Subtract this from the dividend \(x^{3}-9\) to get \(-x-9\). This is the new dividend.
04

Repeat the Division

Repeat steps 2 and 3 for this new dividend. As \(x^{2}\) cannot divide \(-x\), the remainder is \(-x-9\). Now, our long division ends here as the degree of the divisor is more than the degree of the new dividend.
05

Write the Final Answer

Express the final answer as a quotient plus the remainder divided by the divisor. The final answer is \(x-\frac{x+9}{x^{2}+1}\).

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