Problem 32 - Algebra for College Students

Question

Subtract the polynomials using the vertical format. \(3 x-7\) from \(2 x+1\)

Step-by-Step Solution

Answer

The result of the subtraction is \(-x + 8\).

Step 1: Write the Polynomials Vertically

First, write the polynomial expressions vertically, aligning similar terms (terms with the same power of x).The original polynomials given are:1. \(2x + 1\)2. \(3x - 7\)Now we write them vertically:\begin{array}{r}2x + 1 \-(3x - 7) \\hline\end{array}

Step 2: Change Signs of the Second Polynomial

To subtract the polynomials, we must change the signs of the terms in the second polynomial, transforming the subtraction operation into addition:\(2x + 1\) from \(3x - 7\) becomes:1. Change \(3x\) to \(-3x\)2. Change \(-7\) to \(+7\)The vertical alignment now becomes:\begin{array}{r}2x + 1 \+(-3x + 7) \\hline\end{array}

Step 3: Add the Polynomials

Now add the aligned expressions by adding the coefficients of similar terms:1. Add the x-terms: \(2x + (-3x) = -x\)2. Add the constant terms: \(1 + 7 = 8\)So, the result of subtracting the polynomials is:\(-x + 8\).

Key Concepts

Vertical Alignment in Polynomial Subtraction

When it comes to polynomial subtraction, **vertical alignment** is a crucial first step. Simply put, you line up the polynomials one above the other, ensuring that terms with the same power, or degree, are directly in line with one another. This alignment helps in clearly identifying which terms are to be subtracted or combined later.

• Think of this step like stacking numbers in basic arithmetic to ensure the correct columns align.
• The objective is to make sure that all like terms, those that involve the same power of the variable, match up vertically.

For example, in our exercise, when subtracting the polynomials \(3x - 7\) from \(2x + 1\), we write:

• \(2x + 1\)
• \(3x - 7\)

Ensuring terms with the same degree (e.g., \(x\) terms and constant terms) are aligned is the key benefit of this approach. After aligning, subtraction becomes straightforward.

Coefficient Addition

Once you've aligned the polynomials vertically, the next step in subtraction involves working with the **coefficients**. But before we add or subtract, we must flip the signs of the coefficients in the polynomial being subtracted. This turns the subtraction operation into an addition problem, which is often easier to conceptualize.

• For example, change the sign of each term in the second polynomial.
• \(3x\) becomes \(-3x\) and \(-7\) becomes \(+7\).

Now, perform the calculation to combine terms:

• Add the coefficients of the \(x\) terms: \(2x + (-3x) = -x\)
• Add the constant terms: \(1 + 7 = 8\)

By focusing on the coefficients and turning subtraction into addition, the process becomes simpler and reduces the risk of errors.

Grouping Same Power Terms

In polynomial subtraction, dealing with **same power terms** efficiently is crucial. These are terms that have the same variable raised to the same power. When vertically aligned, it's easy to group such terms and perform the necessary operations.

• The concept is similar to sorting like items in arithmetic.
• For example, both \(2x\) and \(-3x\) are grouped because they are terms with \(x^1\), and \(1\) and \(7\) are grouped as constants.

When subtracting, it is these groupings that allow for straightforward calculations. Collectively, this makes the entire operation smooth:

• Subtract or add the coefficients of \(x\)-terms together.
• Likewise, combine the constants.

This approach ensures clarity and simplicity in polynomial operations, making the process less intimidating and more accessible.