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Chapter 5: Problem 29

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{27 x^{3}-1}{3 x-1}$$

Short Answer

Expert verified
The quotient is \(9x^{2}+3x\) and the remainder is \(-1\)

Step by step solution

01

Perform the Division

Divide the first term of the dividend \(27x^{3}\) by the first term of the divisor \(3x\), which yields \(9x^{2}\). Then, multiply the entire divisor \(3x-1\) by the obtained quotient \(9x^{2}\) obtaining \(27x^{3} - 9x^{2}\). After that, subtract the result of that multiplication from the dividend. So the updated dividend now is \(-(27x^{3}-9x^{2})+(27x^{3}-1) = 9x^{2} - 1\).
02

Repeat the Division

Now, divide the first term of the updated dividend \(9x^{2}\) by the first term of the divisor \(3x\) giving \(3x\). Do the same procedure again, multiply the divisor by the new quotient \(3x\) and subtract it from the updated dividend. So, the last dividend is \(-(9x^{2} - 3x)+(9x^{2} - 1) = 3x - 1\). Obviously that process won't continue more since the degree of the updated dividend is less than the divisor's degree.
03

Combine the Quotients

Add the two obtained quotients, which results in \(9x^{2}+3x\). Therefore, the quotient of the division is \(9x^{2}+3x\). The remainder of the division, noted as it is the last dividend, is \(-1\).
04

Check the Result

Multiplying the quotient \((9x^{2}+3x)\) by the divisor \((3x-1)\) and adding the remainder \(-1\), it should result in the original dividend \(27x^{3}-1\). Therefore, \((9x^{2}+3x)(3x-1) -1 = 27x^{3} -1\). Hence, the answer is verified.

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