Problem 62 - Beginning and Intermediate Algebra

Question

Do you prefer adding and subtracting polynomials vertically or horizontally? Why?

Step-by-Step Solution

Answer

I prefer adding and subtracting polynomials vertically, as it makes it easier to see like terms and prevents mistakes when adding or subtracting coefficients. However, it requires more space compared to the horizontal method, which can be visually simpler but also easier to miss like terms and make mistakes. Ultimately, the preferred method depends on personal preferences and understanding of the process.

Step 1: 1. Vertical Addition and Subtraction of Polynomials

In the vertical method, align the polynomials so that like terms (terms with the same degree) are in the same column: 3x^2 + 2x - 4 + x^2 - 3x + 2 ------------------------------------------

Step 2: 2. Perform the Operations

Now, we can either add or subtract the like terms: For addition: + (3x^2 + x^2) + (2x - 3x) + (-4 + 2) + 4x^2 + (-1x) + (-2) So, P(x) + Q(x) = 4x^2 - x - 2 For subtraction: + (3x^2 - x^2) + (2x + 3x) - (-4 - 2) + 2x^2 + (5x) + (-6) So, P(x) - Q(x) = 2x^2 + 5x - 6

Step 3: 3. Horizontal Addition and Subtraction of Polynomials

In the horizontal method, write the polynomials in a row: For addition: P(x) + Q(x) = (3x^2 + 2x - 4) + (x^2 - 3x + 2) For subtraction: P(x) - Q(x) = (3x^2 + 2x - 4) - (x^2 - 3x + 2)

Step 4: 4. Perform the Operations

Now, combine the like terms: For addition: P(x) + Q(x) = 3x^2 + 2x - 4 + x^2 - 3x + 2 = 4x^2 - x - 2 For subtraction: First, distribute the negative sign: P(x) - Q(x) = 3x^2 + 2x - 4 - x^2 + 3x - 2 Now, combine like terms: P(x) - Q(x) = 2x^2 + 5x - 6 Having explained both methods, let's compare their pros and cons: - Vertical Method: Pros: Easier to see like terms and prevent mistakes when adding or subtracting coefficients. Cons: Requires more space. - Horizontal Method: Pros: Uses less space and can be visually simpler. Cons: Easier to miss like terms and make mistakes when adding or subtracting coefficients. Now that you understand the differences between adding and subtracting polynomials vertically and horizontally, you can decide which method you prefer based on your personal preferences and how you best understand the process.