Problem 31 - Algebra for College Students

Question

Subtract the polynomials using the vertical format. \(5 x-2\) from \(12 x+6\)

Step-by-Step Solution

Answer

The result is \(7x + 8\).

Step 1: Arrange Vertically

Write the two polynomials vertically, aligning the like terms. The polynomial to be subtracted, \(5x - 2\), is written below \(12x + 6\). It looks like this:\[\begin{array}{r}12x + 6 \- (5x - 2)\end{array}\]

Step 2: Subtract Coefficients of Like Terms

Subtract the coefficients of the \(x\) terms and the constant terms separately. Start from the left of the polynomials:- For the \(x\) terms: \(12x\) minus \(5x\) equals \(7x\).- For the constant terms: \(6\) minus \(-2\) equals \(8\) because subtracting a negative is the same as adding.The result is:\(7x + 8\).

Step 3: Write the Final Expression

Combine the results from Step 2 to write the final expression as:\[7x + 8\]

Key Concepts

Vertical Format

When subtracting polynomials, the vertical format is a convenient way to organize and visually process the equation. Just like adding large numbers, arranging polynomials in vertical format helps keep track of like terms more accurately. In this process, you align the polynomials vertically. Make sure each term lines up, so variables and powers match correctly.

In our example, we have:

• The polynomial, \(12x + 6\), on top.
• The polynomial, \(5x - 2\), directly below it.

Now, subtract one polynomial from the other. By organizing it vertically, you're less likely to miss a term or make a mistake when handling the coefficients.

Like Terms

"Like terms" are terms within polynomials that have the same variable raised to the same power. In subtraction, always focus on these terms, as they are the only ones that can be directly combined by addition or subtraction.

For example, in the equation:

• The terms \(12x\) and \(5x\) are like terms because both contain \(x\) raised to the first power.
• Constant numbers, like \(6\) and \(-2\), are also like terms.

Identifying like terms correctly ensures that you only combine terms that are mathematically compatible. This avoids errors and simplifies the expression efficiently.

Subtracting Coefficients

After like terms are aligned, the next step is to subtract their coefficients. Coefficients are the numerical parts in front of the variables. The key here is to deal with each pair of coefficients separately.

In our example:

• For terms with \(x\), subtract \(5\) from \(12\), giving you \(7x\).
• For constant terms, subtract \(-2\) from \(6\). Remember that subtracting a negative is equivalent to adding, resulting in \(6 + 2 = 8\).

This gives you a new polynomial, \(7x + 8\). Always double-check each step to ensure accuracy, especially when dealing with negative numbers.

Negative Numbers in Subtraction

Working with negative numbers in subtraction can be tricky, but it becomes simple once you grasp the rule that subtracting a negative number is like adding its absolute value. This is crucial when we get to terms like the constant \(-2\) in our example.

Think about the expression "minus a negative" as "plus the positive." So when subtracting

• \(-2\) from \(6\), you are actually performing \(6 + 2 = 8\).

Keeping this principle in mind helps avoid common pitfalls when subtracting polynomials, ensuring that the final result is correct and easily understandable.