Problem 6 - Algebra 1

Question

Set up the long division problem, but do not perform the division. Divide \(-x^{2}-4 x+21\) by \(-x+3\)

Step-by-Step Solution

Answer

The setup for long division of \(-x^{2}-4 x+21\) by \(-x+3\) is: \[(-x + 3) \, |\, (-x^2 - 4x + 21) \]

Step 1: Arrange the Dividend and Divisor

Arrange the polynomial \(-x^{2}-4 x + 21\) and the divisor \(-x + 3\) in the format applicable for polynomial division. Remember to write the terms in decreasing powers of \(x\). Ensure that the divisor is also written in the correct order. The division would look this way: \[(-x + 3) \, |\, (-x^2 - 4x + 21) \]

Step 2: Include Placeholder Terms for Missing Powers of x

In polynomial division, if any terms for a certain power of \(x\) are missing from the dividend, we typically include those terms with a coefficient of 0. In this example, every power of \(x\) from the largest in the divisor to 0 is accounted for in the dividend, so this step does not require any action.

Key Concepts

Dividend and Divisor Arrangement

Arranging the dividend and the divisor correctly is a critical first step in polynomial long division. This process is analogous to the way we line up numbers one above the other when dividing numerals. Crucially, the dividend, which is the polynomial we're dividing into, should be written first, followed by the divisor, the polynomial we're dividing by. In our exercise, the dividend is event, and significant care is needed to maintain arithmetic accuracy.

Decreasing Powers of X

When setting up polynomial long division, it's important to arrange both the dividend and the divisor in order of decreasing powers of x. This means you start with the term with the highest exponent and proceed to the one with the lowest. For instance, a polynomial like Here are some reasons why we follow this convention:

• Consistency in Division: Arranging the powers in descending order creates a uniform approach, which is essential in polynomial division.
• Ease of Calculation: It's easier to align similar terms and subtract them when they are organized in a consistent manner.
• Error Prevention: This order prevents the misalignment of terms, which can lead to calculation errors.

Keeping our terms in descending powers helps us see the 'big picture' of the polynomial and proceed systematically during division.

Placeholder Terms in Polynomial Division

In some instances, our polynomials might skip certain powers of x. When this happens, it is important to include placeholder terms with a coefficient of zero to fill the gaps. This ensures that each descending power of x is represented and has a corresponding position in the long division setup.

Why Use Placeholders?

Placeholders maintain the structure of the polynomial, which is particularly important as each step in long division aligns with the degrees of the terms involved. They also serve as visual reminders of the terms we 'carry down' during the division process. Although in the exercise provided, every power of x was accounted for and placeholders were not necessary, including them when required is crucial for keeping the division orderly and ensuring no term is accidentally overlooked.