Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Use long division to divide. Divisor \(x^{2}-1\) Dividend $$x^{3}-27$$

Step-by-Step Solution

Answer

The final answer then becomes \(x\) with the remainder \(-27\). This means \(x^{3}-27\) divided by \(x^{2}-1\) equals \(x\) with a remainder of \(-27\).

Step 1: Setting up

Set up the long division, putting the dividend \(x^{3}-27\) under the division symbol and the divisor \(x^{2}-1\) to the left.

Step 2: First Division

To start with, divide the leading term of the dividend \(x^{3}\) by the leading term of the divisor \(x^{2}\) to get \(x\). This becomes the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the divisor by the first term of the quotient and subtract this from the dividend. This gives us \(-(x^{3} - (x \cdot x^{2} - x)) = -27 \) as the result.

Step 4: Second Division

Next, divide the leading term of the result by the leading term of the divisor again. This time, since the result is \(-27\) and the divisor is \(x^{2} - 1\), which are incommensurable, there are no more terms to add to the quotient.

Step 5: Write down the remaining terms

As there are no further terms to add, take the remaining terms as the rest. The rest is \(-27\) which makes it the final answer, since we can't subtract anything else from it.

Key Concepts

Long Division

Long division is a method used to divide two numbers or expressions, where one is the divisor and the other is the dividend.
For polynomial division, this method involves several repetitive steps of division, multiplication, and subtraction, very much like long division with numbers. It helps to break down complex polynomials into simpler terms or find out how many times a polynomial fits into another.

• When applied to polynomials, it helps identify a quotient and potentially a remainder.
• The process continues until all terms being divided are handled.

The systematic nature of long division makes it a powerful tool in algebra, allowing us to simplify expressions that might seem complex at first glance.

Quotient

The quotient is the result obtained from dividing one polynomial by another using long division. In our exercise:

• The first step is taking the leading term from the dividend and dividing it by the leading term of the divisor.
• This result gives the first term of the quotient.

Continuing the steps of division, multiplication, and subtraction eventually yields a complete polynomial as the quotient, or it shows that the division resulted in a remainder.

Divisor

The divisor is the polynomial that divides the dividend. It is the expression outside the long division symbol.
In our specific exercise, the divisor is \(x^2 - 1\). It plays a crucial role as dividing begins by comparing the highest degree term of the divisor and the dividend:

• The degrees of these terms guide the initial division step.
• The structure of the divisor can indicate possible factors or zeros of the dividend when division is complete.

The divisor directly influences what terms will appear in the quotient, as it's repeatedly multiplied by emerging terms in the quotient.

Dividend

In a long division setup, the dividend is the polynomial placed under the division symbol. It is the expression being divided by the divisor. In the exercise at hand, the dividend is \(x^3 - 27\). Typically:

• The division begins by focusing on the leading (highest degree) term of the dividend.
• The rest of the terms are addressed step by step as we progress through the polynomial long division.

Through each step, the dividend is reduced by an amount calculated using the divisor and the emerging terms in the quotient until it possibly becomes zero, or a remainder is left.