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Chapter 5: Problem 49

In your own words, explain how to divide a polynomial by a binomial. Use \(\frac{x^{2}+4}{x+2}\) in your explanation.

Short Answer

Expert verified
The quotient of the polynomial \(x^{2}+4\) divided by the binomial \(x+2\) is \(x\), with a remainder of -2x + 4 which can be expressed as \(\frac{-2x+4}{x+2}\).

Step by step solution

01

Set Up the Long Division

First, set up the division using long division style. \(x+2\) will be the divisor outside, and \(x^{2}+4\) will be the dividend inside.
02

Divide the first term

Take the leading term from the divisor \(x\) and divide the first term of the divident \(x^{2}\) by it. The result is \(x\) which will be the first term of the quotient.
03

Multiply and Subtract

Multiply the divisor \(x+2\) by the first term of the quotient, \(x\), to get \(x^{2}+2x\). Subtract this from the dividend \(x^{2}+4\). The result after simplifying (\(x^{2} - (x^{2} + 2x)\)) is -2x + 4.
04

Repeat the process

-2x + 4 is now the new dividend. But notice that its degree is less than the divisor鈥檚 degree and so we stop here. This is the remainder of the division.
05

Present the final answer

The division of the polynomial \(x^{2} + 4\) by the binomial \(x+2\) results in the quotient \(x\) with a remainder of -2x + 4. This result can also be expressed as \(\frac{-2x+4}{x+2}\) after the leading term.

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