Problem 24 - Algebra for College Students

Question

Subtract the polynomials using the horizontal format. \(5 a+7\) from \(-a-4\)

Step-by-Step Solution

Answer

The result of the subtraction is \(-6a - 11\).

Step 1: Write the Expression to Subtract

The problem asks us to subtract the polynomial \(5a + 7\) from \(-a - 4\). First, let's write this subtraction operation explicitly: \((-a - 4) - (5a + 7)\).

Step 2: Distribute the Negative Sign

Subtracting a polynomial is the same as adding its opposite. Distribute the negative sign across \(5a + 7\) to get \((-a - 4) - 5a - 7\). This gives us: \(-a - 4 - 5a - 7\).

Step 3: Combine Like Terms

Now, combine like terms from the expression \(-a - 4 - 5a - 7\). The \(-a - 5a\) terms combine to \(-6a\), and the \(-4 - 7\) terms combine to \(-11\).

Step 4: Write the Resulting Polynomial

Combine the results from the previous step to get the final polynomial expression. The resulting polynomial after subtraction is \(-6a - 11\).

Key Concepts

Combining Like Terms

When dealing with polynomials, it’s essential to combine like terms to simplify your expressions. Like terms are terms that have the same variables raised to the same power. For example, in the exercise, both \(-a\) and \(-5a\) are like terms because they both have the variable \(a\) raised to the power of 1. Meanwhile, the numbers \(-4\) and \(-7\) are also like terms as they are constant terms, meaning they have no variables attached.

The process of combining like terms involves adding or subtracting the coefficients (the numbers in front of the variables) of these like terms. Here's what happens in our exercise:

• Combine \(-a\) and \(-5a\) to get \(-6a\). This is done by adding the coefficients, -1 and -5, to get -6.
• Combine the constants \(-4\) and \(-7\) to get \(-11\). Again, this is done by simply adding -4 and -7, resulting in -11.

By combining like terms, we make expressions simpler and easier to work with, which is a crucial step in solving algebraic problems.

Distribution in Algebra

Distribution is a key operation in algebra that involves spreading one term across others, usually within parentheses, by multiplication. Specifically, when you subtract a polynomial, you are essentially adding its opposite. This requires distributing the negative sign across all terms in the second polynomial.

In the exercise, the subtraction step looks like this:

• Original expression: \((-a - 4) - (5a + 7)\).
• Apply the negative sign: The negative sign in front of \((5a + 7)\) means you need to take the opposite of each term within the parentheses. This changes \(5a + 7\) to \(-5a\) and \(-7\).
• The expression becomes \(-a - 4 - 5a - 7\) once the distribution is complete.

Remember that distribution helps to ensure that every term in a set of parentheses is properly accounted for when conducting arithmetic operations. It is particularly important in subtraction to maintain the correct signs.

Negative Sign in Polynomials

Understanding how negative signs operate within polynomials is crucial for accurate problem-solving in algebra. Subtracting a polynomial can change the sign of each term within it, which is why it's important to handle these signs correctly.

Here's how the negative sign plays a role in the given exercise:

• The expression starts as \((-a - 4) - (5a + 7)\). When the negative sign is distributed, it affects both \(5a\) and \(7\), turning them into \(-5a\) and \(-7\).
• This sign change is necessary because subtracting a positive is equivalent to adding a negative.
• Failure to correctly apply the negative sign can lead to incorrect combinations and an incorrect final expression.

Always be cautious about the placement and effect of negative signs in algebraic expressions; they influence the direction of your arithmetic and are key to achieving the right answer.