Question

Do you prefer adding two polynomials using the horizontal method or the vertical method? Write a brief position paper defending your choice.

Short answer

Prefer the horizontal/vertical method due to its simplicity/clarity and reduced error rate.

Step-by-step solution

Understand the Horizontal Method

In the horizontal method, each polynomial is written on the same line. Then, like terms are grouped and added together. For example, if adding polynomials \(p(x) = 2x^2 + 3x + 1\) and \(q(x) = x^2 + 4\), you write them as \(p(x) + q(x) = (2x^2 + 3x + 1) + (x^2 + 4)\) which simplifies to \((2x^2 + x^2) + 3x + (1 + 4) = 3x^2 + 3x + 5\).

Understand the Vertical Method

In the vertical method, polynomials are written in columns, aligning like terms. For example, for the polynomials \(p(x) = 2x^2 + 3x + 1\) and \(q(x) = x^2 + 4\), write them as:\begin{array}{c@{}c@{}c@{}c}&2x^2 &+3x &+1 \ + &x^2 & &+4 \ \text{----------------} \ &3x^2& +3x & +5\text{----------------} \right.

Compare Both Methods

Compare both methods in terms of ease and clarity. Some may find the horizontal method simpler as it directly combines terms in one line, making it easy to follow. Others might prefer the vertical method for the clear alignment of like terms, which can reduce errors especially with more complex expressions.

State Your Position

Decide which method you prefer and justify your choice. If preferring the horizontal method, you might argue that it is more straightforward and less time-consuming. If preferring the vertical method, you might argue that it offers a clearer, more organized way to combine like terms, reducing the chance of mistakes.

Key concepts

Horizontal Method

The horizontal method for adding polynomials involves writing the expressions on a single line. This method makes it easy to group and combine like terms. For example, let's consider the polynomials \(p(x) = 2x^2 + 3x + 1\) and \(q(x) = x^2 + 4\). In the horizontal method, you would write them as:
\((2x^2 + 3x + 1) + (x^2 + 4)\)
Next, you group the like terms together and add them step by step:
\((2x^2 + x^2) + 3x + (1 + 4)\)
This simplifies to:
\(3x^2 + 3x + 5\).

The horizontal method is often preferred for its simplicity and straightforwardness. All terms are combined in one go, making it quick and easy to follow.

Vertical Method

The vertical method involves writing the polynomials in columns, lining up like terms. This approach can be particularly useful for more complex expressions as it provides a clearer view of each term. For example, take \(p(x) = 2x^2 + 3x + 1\) and \(q(x) = x^2 + 4\). You would write them vertically:

• \(2x^2 + 3x + 1\)
• \( + x^2 + 0x + 4\)

Aligning like terms (note the placeholder \(0x\) to match the terms):

\begin{array}{c@{}c@{}c@{}c}&2x^2 &+3x &+1 \ + &x^2 & &+4 \ \text{----------------} \br)3x^2 & +3x & +5 \ \text{----------------}
The vertical method makes the addition process more organized. By clearly aligning the terms, you can reduce the chance of mistakes and ensure that all terms are correctly added.

Like Terms

Like terms are terms that have the same variable raised to the same power. Identifying and grouping like terms is a crucial step in polynomial addition. For example, in the polynomials \(2x^2 + 3x + 1\) and \(x^2 + 4\), the like terms are:

• \(2x^2\) and \(x^2\)
• \(3x\) (note that there is no \(x\) term in the second polynomial)
• \(1\) and \(4\) (constants)

To add like terms, simply combine their coefficients. Using the horizontal method:
\((2x^2 + x^2) + 3x + (1 + 4)\) yields:
\(3x^2 + 3x + 5\)

Properly identifying and grouping like terms ensures that all parts of the polynomial are correctly combined.

Polynomial Addition

Polynomial addition is the process of combining polynomials by adding the like terms. This operation is fundamental in algebra and is used in various mathematical contexts. The two main methods for adding polynomials are the horizontal method and the vertical method. Here is a step-by-step guide to polynomial addition:

• Write down the polynomials to be added.
• Identify and group the like terms.
• Combine the coefficients of the like terms.
• Simplify the result to get the final polynomial.

For example, adding \(p(x) = 2x^2 + 3x + 1\) and \(q(x) = x^2 + 4\) using the horizontal method:
\((2x^2 + 3x + 1) + (x^2 + 4)\) simplifies to:
\(3x^2 + 3x + 5\)

The result is a new polynomial obtained by adding the coefficients of the like terms.

Mathematical Methods Comparison

When comparing the horizontal and vertical methods for polynomial addition, it's essential to consider ease, clarity, and risk of errors. Both methods have their advantages:

• Horizontal Method: This method is straightforward and quick, combining all terms in one line. It's suitable for simpler polynomial expressions as it reduces the number of steps.


• Vertical Method: This method provides a clearer structure by aligning like terms in columns. It minimizes the risk of errors, particularly with complex expressions, as each term's place is clearly defined.

Your choice between these methods may depend on the specific problem and your comfort with organizing terms. No matter the method, the goal is to combine like terms accurately to achieve the correct polynomial result.