Problem 49 - Elementary and Intermediate Algebra

Question

Subtract the polynomials. $$ \left(3 a^{2}-2 a+4\right)-\left(a^{2}-3 a+7\right) $$

Step-by-Step Solution

Answer

The result is \(2a^2 + a - 3\).

Step 1: Write Down the Problem

We have the expression \((3a^2 - 2a + 4) - (a^2 - 3a + 7)\). This is a subtraction of two polynomials.

Step 2: Distribute the Negative Sign

The negative sign outside the second polynomial means we must change the sign of each term within the polynomial. This gives us: \((3a^2 - 2a + 4) - a^2 + 3a - 7\).

Step 3: Combine Like Terms

We combine the terms with the same powers of \(a\): \((3a^2 - a^2) + (-2a + 3a) + (4 - 7)\). This simplifies to \(2a^2 + a - 3\).

Key Concepts

Distributing the Negative Sign in Polynomial Subtraction

When you encounter a subtraction problem involving polynomials, one of the most crucial steps is to distribute the negative sign across the terms of the polynomial being subtracted. It might sound a bit technical, but it simply means changing the sign of each term in the second polynomial to its opposite.
For example, consider the expression \[ (3a^2 - 2a + 4) - (a^2 - 3a + 7) \]The negative sign in front of the second polynomial—\((a^2 - 3a + 7)\)—requires us to change each of its terms. We do this by converting each term as follows:

• Turn \(a^2\) into \(-a^2\)
• Turn \(-3a\) into \(+3a\)
• Turn \(+7\) into \(-7\)

After distributing the negative sign, the expression becomes:\[ 3a^2 - 2a + 4 - a^2 + 3a - 7 \]This is a foundational skill in polynomial subtraction and ensures that all terms are correctly adjusted.

Combining Like Terms in Polynomial Subtraction

After distributing the negative sign, the next key concept is combining like terms. This step simplifies the expression by collecting terms with the same degree of the variable together. Here, we focus on pairing terms that share the same power of the variable, such as terms involving \(a^2\), \(a\), etc.
Let's look at our adjusted expression:\[ 3a^2 - a^2 - 2a + 3a + 4 - 7 \]To combine like terms, follow these easy steps:

• Identify and group the \(a^2\) terms: \(3a^2 - a^2\)
• Identify and group the \(a\) terms: \(-2a + 3a\)
• Identify and group the constant terms: \(4 - 7\)

Solving each group gives us:

• \(3a^2 - a^2 = 2a^2\)
• \(-2a + 3a = a\)
• \(4 - 7 = -3\)

These simplifications lead to the final expression:\[ 2a^2 + a - 3 \]Combining like terms is essential to making polynomial expressions more straightforward.

Understanding Polynomial Subtraction

Subtracting polynomials involves several intuitive steps, but with practice, it becomes much more manageable. Unlike simple arithmetic subtraction, this process requires handling terms according to their structure and degree.
Let's recap the process using our example:We began with:\[ (3a^2 - 2a + 4) - (a^2 - 3a + 7) \]The aim was to address each component of the polynomial subtraction:1. **Distribute the Negative Sign:** Adjust terms in the polynomial being subtracted to their opposite forms: \( -a^2, +3a, -7 \)2. **Combine Like Terms:** Align and simplify terms sharing the same degree: \( 2a^2, a, -3 \)This leads us to the final simplified polynomial:\[ 2a^2 + a - 3 \]Embracing the methodical process of polynomial subtraction is crucial. It ensures expressions are handled accurately and efficiently. Once mastered, these polynomials become less intimidating, laying the foundation for more advanced algebraic operations.