Problem 22 - Algebra for College Students

Question

Subtract the polynomials using the horizontal format. \(7 x+5\) from \(2 x-1\)

Step-by-Step Solution

Answer

The result is \(-5x - 6\).

Step 1: Identify the Polynomials

We have two polynomials involved in this subtraction problem: \(7x + 5\) and \(2x - 1\). The task requires subtracting the polynomial \(7x + 5\) from \(2x - 1\).

Step 2: Set Up the Subtraction Expression

Rewrite the subtraction problem in horizontal format: \((2x - 1) - (7x + 5)\). This expression clearly indicates that we need to subtract each term of \(7x + 5\) from \(2x - 1\).

Step 3: Distribute the Negative Sign

Apply the negative sign to each term in the second polynomial: \((2x - 1) - 7x - 5\) becomes \(2x - 1 - 7x - 5\). This step ensures each term of the subtracted polynomial is considered correctly.

Step 4: Combine Like Terms

Start by combining the \(x\) terms: \(2x - 7x = -5x\). Next, combine the constant terms: \(-1 - 5 = -6\).

Step 5: Write the Final Solution

The result after combining like terms is \(-5x - 6\). This is the polynomial after performing the subtraction.

Key Concepts

Horizontal Format

When dealing with polynomial subtraction, the horizontal format is a handy way to set up the problem. It allows you to see each part of the operation laid out in a single line. This format is especially useful when subtracting one polynomial from another, as it directly shows which terms will interact. For example, consider subtracting the polynomial \(7x + 5\) from \(2x - 1\). You would write the expression as

• \((2x - 1) - (7x + 5)\)

This presentation makes it easier to handle subsequent steps, such as distributing negative signs and combining like terms, all visible at a glance.Seeing everything in one line can help you better understand which terms go together and what operations need to be performed. This clarity is one of the main advantages of using the horizontal format for polynomial subtraction.

Combine Like Terms

After you have set up your polynomials in the horizontal format, it's time to combine like terms. Like terms are terms with the same variable raised to the same power. To effectively combine them, you first perform any necessary arithmetic across similar terms.For our example, after distributing the negative sign, we have the expression:

• \(2x - 1 - 7x - 5\)

Next, you identify and combine the like terms:

• \(x\) terms: \(2x\) and \(-7x\) are like terms.
• Constants: \(-1\) and \(-5\) are also like terms.

By combining these, you get

• \(2x - 7x = -5x\)
• \(-1 - 5 = -6\)

Finally, put together these simplified terms to obtain the polynomial \(-5x - 6\). Successfully combining like terms is crucial for presenting the simplest form of your solution.

Distribute the Negative Sign

Distribution of the negative sign is a pivotal step in polynomial subtraction. This ensures that the subtraction operation affects each term of the polynomial being subtracted. Without proper distribution, errors in calculations are likely.In our exercise, after setting the problem as

• \((2x - 1) - (7x + 5)\)

we have to distribute the negative sign across the polynomial to be subtracted:

• \((2x - 1) - 7x - 5\)

This transforms the expression into

• \(2x - 1 - 7x - 5\)

Notice how the negative sign will change the operation for each term in \(7x + 5\). This simplification is crucial as it sets up the problem correctly for combining like terms.