Problem 34 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The result is \(11x + 8\).

Step 1: Arrange the Polynomials

Write the polynomials one above the other, ensuring that like terms are aligned vertically. Place the polynomial you are subtracting underneath:\[ \begin{array}{c} 5x + 6 \- (-6x - 2) \end{array} \]

Step 2: Distribute the Negative Sign

Remember to change the sign of each term in the polynomial being subtracted. This is because subtracting a number is the same as adding its opposite:\[ \begin{array}{c} 5x + 6 \+ 6x + 2 \\hline\end{array} \]

Step 3: Add the Polynomials

Now that both polynomials are organized with like terms, add them together. Combine the coefficients of like terms:\[ \begin{array}{c} (5x + 6x) + (6 + 2) \\hline11x + 8 \end{array} \]

Key Concepts

Vertical Format in Polynomial Subtraction

When subtracting polynomials, using the vertical format is like making a list of what you have. It's a bit like adding numbers vertically. To do this with polynomials, you need to line them up correctly. Here's how to do it:

• Start by writing one polynomial above the other.
• Align the terms with similar variables and exponents. This is crucial for simplifying later on.
• Place the polynomial you are subtracting underneath.

For example, if you want to subtract \(-6x - 2\) from \(5x + 6\), you write them as:\[\begin{array}{c}5x + 6 \-(-6x - 2)\end{array} \]This method helps you see exactly which terms you are working with and ensures nothing gets left out.

Understanding Like Terms

Identifying like terms is key when working with polynomials. Like terms are those that have the same variable raised to the same power. Only coefficients of like terms can be combined because their variable part is identical. Here's what to remember:

• Look for terms with exactly the same variable parts.
• It's only the numerical coefficients that change when you add or subtract.
• Write like terms directly beneath each other in the vertical format.

In our example, the like terms were \(5x\) and \(-6x\), both having the variable \(x^1\), and the constants \(6\) and \(-2\). By organizing the terms, you can easily combine \(5x + 6x\) and \(6 + 2\) to get a simplified expression.

Distributing the Negative Sign

Distributing the negative sign is crucial in polynomial subtraction, as it ensures accurate results. Here's why this step is essential: in mathematics, subtracting something is the same as adding its opposite. When you see a negative sign in front of parentheses, you must change the signs of every term inside those parentheses.

• For example, applying the negative sign in subtracting \(-6x - 2\) turns it into \(+6x + 2\).
• This means you addition can take place, which makes subtraction feel like addition.

Without this step, our calculations would lead to incorrect answers. Once the signs are changed, the process becomes a simple addition: \(5x + 6\) now becomes \(5x + 6x + 6 + 2\), leading to the final answer. It's this flip of signs that keeps everything balanced and correct.