Problem 33 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. \(\left(\frac{1}{2} k^{2}+3\right)\left(12 k^{2}+5 k-10\right)\)

Step-by-Step Solution

Answer

The simplified result of the given expression is \(6k^4 + \frac{5}{2}k^3 + 31k^2 + 15k - 30\).

Step 1: Distribute the terms in the first polynomial

To start, we will distribute each term in the first polynomial, \(\frac{1}{2}k^2 + 3\), by every term in the second polynomial, \(12k^2 + 5k - 10\). We will perform four multiplications in total.

Step 2: Multiply the first term in the first polynomial by all terms in the second polynomial

First, multiply the term \(\frac{1}{2}k^2\) by each term in the second polynomial: \(\frac{1}{2}k^2 \cdot 12k^2 = 6k^4\) \(\frac{1}{2}k^2 \cdot 5k = \frac{5}{2}k^3\) \(\frac{1}{2}k^2 \cdot (-10) = -5k^2\)

Step 3: Multiply the second term in the first polynomial by all terms in the second polynomial

Now, multiply the term \(3\) by each term in the second polynomial: \(3 \cdot 12k^2 = 36k^2\) \(3 \cdot 5k = 15k\) \(3 \cdot (-10) = -30\)

Step 4: Combine the results of the multiplications

Combine the eight terms we obtained in steps 2 and 3: \(6k^4 + \frac{5}{2}k^3 - 5k^2 + 36k^2 + 15k - 30\)

Step 5: Simplify the result

Combine like terms to simplify the expression: \(6k^4 + \frac{5}{2}k^3 + 31k^2 + 15k - 30\) The simplified result, after performing the indicated operations, is: \(6k^4 + \frac{5}{2}k^3 + 31k^2 + 15k - 30\)

Key Concepts

Distributive Property

The distributive property is fundamental in algebra, allowing you to multiply a single term across terms inside a set of parentheses. An easy way to remember this is: "multiply each term by every term." This property lets us break down complex expressions into manageable parts.

In this exercise, we start by distributing each term from the first polynomial, \(\frac{1}{2} k^2 + 3\), across each term in the second polynomial, \(12 k^2 + 5 k - 10\). Here's how it works:

• Multiply \(\frac{1}{2}k^2\) with every term in the second polynomial, resulting in three separate expressions.
• Then, multiply the constant \(3\) with each term in the second polynomial, again producing three expressions.
• Finally, combine both sets of results.

Breaking it down makes it easier to handle larger expressions efficiently.

Simplification of Expressions

Simplification is all about making expressions easier to understand and use. Once we've multiplied and distributed terms, we often end up with many pieces that need to be combined.

Start by writing down each term you get from multiplication. In our example, the multiplication results in eight terms. Keeping these organized is key:

• List all terms from the distribution.
• Look for terms that can be combined (more on this ahead).
• Write the expression in an orderly fashion.

Doing this brings clarity and readiness for the next step—combining like terms, which leads to full simplification.

Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. Identifying them is crucial for combining and simplifying expressions:

Let's revisit our simplified expression: \(6k^4 + \frac{5}{2}k^3 - 5k^2 + 36k^2 + 15k - 30\). Here's how you simplify:

• Collect all terms with the same variable and exponent, such as \(-5k^2\) and \(36k^2\).
• Add or subtract these terms together: \(-5k^2 + 36k^2 = 31k^2\).
• Repeat this for other sets of like terms if needed.

This step transforms an expression into a cleaner, more compact form, making it easier to work with.