Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, find the sum or difference. \(\left(7 a^{3}+6 a^{2}-4 a-13\right)+\left(-3 a^{3}-4 a^{2}+6 a+17\right)\)

Step-by-Step Solution

Answer

The sum is \(4a^3 + 2a^2 + 2a + 4\).

Step 1: Identify Like Terms

To start, identify the like terms in both expressions: these are terms that have the same variable part raised to the same power. In this case, group the terms based on their powers of \(a\): \(a^3\), \(a^2\), \(a\), and the constant terms.

Step 2: Combine Cubic Terms

Add the coefficients of the \(a^3\) terms from both expressions: \(7a^3 - 3a^3 = 4a^3\).

Step 3: Combine Quadratic Terms

Add the coefficients of the \(a^2\) terms: \(6a^2 - 4a^2 = 2a^2\).

Step 4: Combine Linear Terms

Add the coefficients of the \(a\) terms: \(-4a + 6a = 2a\).

Step 5: Combine Constant Terms

Add the constant terms: \(-13 + 17 = 4\).

Step 6: Write the Simplified Expression

Combine all the sums from the previous steps to get the simplified expression: \(4a^3 + 2a^2 + 2a + 4\).

Key Concepts

Understanding Like Terms

In algebra, like terms are key to simplifying expressions and solving equations. Like terms have the same variable with the exact same power. For example, in the expression given in the exercise, terms like \(7a^3\) and \(-3a^3\) are considered like terms because they both contain the variable \(a\) raised to the same power, which is 3. Similarly, \(6a^2\) and \(-4a^2\) are like terms. Like terms can be easily combined by adding or subtracting their coefficients. This process is crucial for simplifying expressions and making calculations easier.

• Identify terms with identical variable and exponent pairs.
• Only combine terms that belong in the same group of like terms.
• Use arithmetic operations to add or subtract the coefficients of these terms.

This simplification process leads to a clearer and more concise expression.

Working with Cubic Terms

Cubic terms are those where the variable is raised to the third power. In the given polynomial, the cubic terms are \(7a^3\) and \(-3a^3\). To combine cubic terms, focus on their coefficients: the numbers in front of \(a^3\).
By adding \(7\) and \(-3\), you simplify the expression to \(4a^3\). This step reduces the complexity of the polynomial, making it easier to solve or further simplify.

• Spot the terms with the highest degree, in this case, cubed terms.
• Sum the coefficients of these cubic terms.

This method ensures that polynomial expressions gradually become more manageable.

Simplifying Quadratic Terms

Quadratic terms involve variables raised to the second power. In the exercise example, \(6a^2\) and \(-4a^2\) are the quadratic terms. To simplify them, you add their coefficients, \(6 + (-4)\), resulting in \(2a^2\).
This approach helps in reducing polynomials by grouping middle-degree terms. Quadratic terms are a step down from cubic, focusing on terms with the highest degree after removing cubics.

• Look for terms with the variable squared.
• Add or subtract these terms’ coefficients to find their sum.

Handling quadratic terms properly is essential in polynomial arithmetic.

Breaking Down Linear Terms

Linear terms are the simplest form in polynomials, featuring variables to the power of one, like \(-4a\) and \(6a\) in the activity's expression. Combine them by adding their coefficients, \(-4 + 6\), which gives you \(2a\).
Linear terms are crucial because they represent a straightforward relation involving variables. They help bridge the gap towards constant terms, facilitating the overall simplification of the polynomial.

• Find terms with variables raised to the first power.
• Perform addition or subtraction on the coefficients of these linear terms.
• Ensure correct sign handling during this operation.

Understanding linear terms is foundational for handling any polynomial equations effectively.