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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 division of the polynomials yields a quotient of \(9x^2 + 3x\) which is confirmed and validated by multiplying back with the divisor and proving that it equals the original dividend.

Step by step solution

01

Identify dividend and divisor

Identify the dividend (the expression that is being divided) and the divisor (the expression we are dividing by). The dividend is \(27x^3-1\) and the divisor is \(3x-1\).
02

Start Polynomial Division

Polynomial division is carried out similarly to long division of numbers. The highest degree term of the dividend \(27x^3\) needs to be divided by the highest degree term of the divisor \(3x\). This would give \(9x^2\) as the first part of the quotient.
03

Multiply and Subtract

Multiply the divisor \(3x-1\) by the part of the quotient yielded in the previous step \(9x^2\) and subtract the result from the original dividend \(27x^3-1\). This makes the new dividend: \(27x^3 - (9x^2 * 3x - 9x^2)\) = \(0x^2 + 9x^2\). This new dividend can be written as \(9x^2\). Note that this new dividend has a lesser degree than the initial dividend.
04

Repeat Polynomial Division

Proceed with the polynomial division, treating this new dividend as if it was your original dividend. The highest degree term of the new dividend \(9x^2\) will be divided by the highest degree term of the divisor \(3x\). This gives \(3x\) as the next part of the quotient.
05

Multiply and Subtract Again

Purpose of this step is similar to Step 3. Multiply the divisor \(3x-1\) by the part of the quotient yielded in the previous step \(3x\) and subtract the result from the new dividend \(9x^2\). This makes the resultant second value of the new dividend equal to \(0\). Since we have subtracted all terms of the polynomial, the division process is done.
06

Present Complete Quotient

The complete quotient of the polynomial division is the addition of \(9x^2 + 3x\), giving us \(9x^2 + 3x\) as the final solution.
07

Check the Solution

To check the solution, multiply the divisor \(3x-1\) with the quotient \(9x^2 + 3x\). If the result equals the initial dividend \(27x^3-1\), then the solution is correct. After multiplying and simplifying, the product is found to be equal to the original dividend \(27x^3-1\), proving that the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dividend and Divisor in Algebra
In algebra, performing division with polynomials is quite similar to division with numbers. To comprehend polynomial division, it's crucial to understand the terms 'dividend' and 'divisor'.

The dividend is the polynomial that is to be divided. It is somewhat analogous to the number that we write inside the division box when we perform a simple division. For instance, in the equation \(\frac{27 x^{3}-1}{3 x-1}\), the dividend is \(27x^3-1\).

The divisor, on the other hand, is the polynomial by which the dividend is to be divided. Imagine it as the number outside the division box in elementary division operations. In our example, \(3x-1\) plays the role of the divisor.
Polynomial Division Steps
Dividing polynomials can seem daunting, but breaking it down into steps makes the process manageable.

Setup:

Write the dividend and divisor clearly and identify the highest degree terms.

Divide the Lead Terms:

First divide the highest degree term in the dividend by the highest degree term in the divisor to find the first term of the quotient.

Multiply and Subtract:

Multiply the whole divisor by the new term you've found and subtract it from the dividend, reducing the polynomial's degree.

Repeat:

Continue this process with the new, lower-degree polynomial that results from subtraction.

When you can no longer divide because the dividend's degree is less than the divisor's, you've arrived at a remainder if there's leftover or your division is complete if there's no remainder. Consequently, this systematic strategy simplifies what can appear to be a complicated algebraic operation into a sequence of manageable tasks.
Checking Polynomial Division Solutions
Once you have calculated the quotient, it's important to validate your solution. Always check your work by multiplying the divisor and the quotient together and then adding any remainder. This outcome should be identical to your original dividend.

If you find that your result matches the dividend, then you've done the polynomial long division correctly! If not, retrace your steps to identify and correct the error. Effective checking often helps in correcting mistakes and ensures understanding of the division process, practicing this reaffirms your grasp on the concept and improves your algebraic skills.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$(2 x+3-5 y)(2 x+3+5 y)=4 x^{2}+12 x+9-25 y^{2}$$

You just signed a contract for a new job. The salary for the first year is \(\$ 30,000\) and there is to be a percent increase in your salary each year. The algebraic expression $$\frac{30,000 x^{n}-30,000}{x-1}$$ describes your total salary over n years, where \(x\) is the sum of 1 and the yearly percent increase, expressed as a decimal. a. Use the given expression and write a quotient of polynomials that describes your total salary over three years. b. Simplify the expression in part (a) by performing the division. c. Suppose you are to receive an increase of \(5 \%\) per year. Thus, \(x\) is the sum of 1 and \(0.05,\) or \(1.05 .\) Substitute 1.05 for \(x\) in the expression in part (a) as well as in the simplified form of the expression in part (b). Evaluate each expression. What is your total salary over the three-year period?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Each statement applies to the division problem $$\frac{x^{3}+1}{x+1}$$ The purpose of writing \(x^{3}+1\) as \(x^{3}+0 x^{2}+0 x+1\) is to keep all like terms aligned.

will help you prepare for the material covered in the next section. Use the distributive property to multiply: \(3 x(x+5)\)

determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. I used two points and a checkpoint to graph \(y=x^{2}-4\)
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