Problem 8 - Algebra 1

Question

Set up the long division problem, but do not perform the division. Divide \(72-18 x+x^{2}\) by \(x-6\)

Step-by-Step Solution

Answer

The long division problem set up is \[(x^{2} - 18x + 72) \div (x - 6)\] where the dividend is \(x^{2} - 18x + 72\) and the divisor is \(x - 6\)

Step 1: Arrange the Polynomial

First, arrange the polynomial terms in descending order of power from highest to lowest. Hence, the polynomial becomes \(x^{2} - 18x + 72\).

Step 2: Write the Division Structure

Write down the division structure. The result should look like long division in arithmetic. The dividend \(x^{2} - 18x + 72\) goes on the inside, and the divisor which is \(x - 6\) goes on the outside.

Step 3: Set up Placeholders

Now, write the first term of the divisor (which is 'x') into the division bar. Then, put a placeholder of 0 multiplied by the divisor term 'x' for each power of x that is missing. This problem does not need any placeholders as we have all the required terms.

Step 4: Set-up Only

As the instructions state to 'set up the long division but do not perform the division,' the set-up is complete at this point. Be sure to review to make sure everything is set up correctly for whenever the division should be performed.

Key Concepts

Long Division

Long division is not just for numbers; it can also be used with polynomials. It's a method to divide one expression by another, similar to dividing numbers using long division in arithmetic. To start a polynomial long division, we need to organize everything in a particular format.
Here's how it works:

• Write the dividend (the polynomial to be divided) inside the division bracket.
• Place the divisor (the polynomial you are dividing by) outside the bracket.

The process closely mirrors numerical long division:

• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by this result and write it under the dividend.
• Subtract this product from the dividend to find the remainder.
• Repeat these steps with the new polynomial created from the remainder.

Though we haven’t performed these steps in our exercise, setting up like this prepares us well for when we actually need to perform the division.

Polynomial Structure

Understanding the structure of a polynomial helps us effectively set up long division. Polynomials are expressions consisting of variables raised to whole number powers.
Each term is composed of:

• A coefficient, which is a constant number multiplying the variable.
• The variable, often represented by letters such as 'x' or 'y.'
• An exponent, indicating the power to which the variable is raised.

For any polynomial expression, it's essential to write terms in descending order of their powers—that is, starting with the highest exponent.
In our exercise example, the polynomial to be divided is rearranged as: \[ x^{2} - 18x + 72 \]This rearrangement is crucial for performing long division as it aligns the terms in order, simplifying computation. If any powers of the variable are missing, placeholders (like 0 multiplied by the missing power) help keep track of terms throughout the division process.

Dividend and Divisor

In polynomial division, terms 'dividend' and 'divisor' have specific meanings just like they do in arithmetic division. Knowing which polynomial is the dividend and which one is the divisor helps properly set up the division.

• The **dividend** is the polynomial being divided, written inside the long division bracket. It's analogous to the numerator in a fraction.
• The **divisor** is the polynomial by which we divide, recorded outside the division bracket. It's similar to the denominator in fractions.

For our exercise:- The dividend is \( x^{2} - 18x + 72 \)- The divisor is \( x - 6 \)In algebraic division, just like numeric division, correctly identifying the dividend and divisor is essential for setting up the long division correctly. Each step in division revolves around these two expressions, making it a fundamental concept in polynomial division.