Problem 34 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$\left(\frac{2}{3} c^{2}-8\right)\left(6 c^{2}-4 c+9\right)$$

Step-by-Step Solution

Answer

The simplified expression after performing the indicated operation and combining like terms is: \(\boxed{4c^4 - \frac{8}{3}c^3 - 42c^2 + 32c - 72}\).

Step 1: Identify terms in each binomial

The given expression consists of two binomials: \(\left(\frac{2}{3}c^2 - 8\right)\) and \(\left(6c^2 - 4c + 9\right)\).\newline In the first binomial, the terms are \(\frac{2}{3}c^2\) and \(-8\). In the second binomial, the terms are \(6c^2\), \(-4c\), and \(9\).

Step 2: Multiply each term in the first binomial by each term in the second binomial

Using the distributive property, we will multiply each term in the first binomial by each term in the second binomial: \(\frac{2}{3}c^2 \cdot 6c^2\) = \(\frac{12}{3}c^4\) = \(4c^4\) \(\frac{2}{3}c^2 \cdot (-4c)\) = \(-\frac{8}{3}c^3\) \(\frac{2}{3}c^2 \cdot 9\) = \(\frac{18}{3}c^2\) = \(6c^2\) \(-8 \cdot 6c^2\) = \(-48c^2\) \(-8 \cdot (-4c)\) = \(32c\) \(-8 \cdot 9\) = \(-72\)

Step 3: Combine like terms and simplify the expression

Now, it's time to combine the terms obtained in Step 2: \(4c^4 - \frac{8}{3}c^3 + 6c^2 - 48c^2 + 32c - 72\newline 4c^4 - \frac{8}{3}c^3 - 42c^2 + 32c - 72\) The simplified expression after performing the indicated operation and combining like terms is: \(\boxed{4c^4 - \frac{8}{3}c^3 - 42c^2 + 32c - 72}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to remove parentheses by distributing multiplication over addition or subtraction. This is particularly useful when dealing with polynomials, like in the given exercise.

To apply the distributive property, you take each term in the first binomial, \(\left(\frac{2}{3}c^2 - 8\right)\), and multiply it by each term in the second binomial, \(\left(6c^2 - 4c + 9\right)\). It might seem a bit like a cross-multiplication at first.

• Multiply \(\frac{2}{3}c^2\) by each term in the second binomial.
• Multiply \(-8\) by each term in the second binomial.

This approach ensures every possible multiplication between each term from the two binomials is considered, which lays the foundation for further simplification.

Binomial

A binomial is an algebraic expression that contains exactly two terms. In the exercise, \(\frac{2}{3}c^2 - 8\) is the first binomial. It consists of the terms \(\frac{2}{3}c^2\) and \(-8\).

The second expression, \(6c^2 - 4c + 9\), though it is actually a trinomial (having three terms), was mentioned as part of the multiplication with the first binomial. These terms include \(6c^2\), \(-4c\), and \(9\).

Understanding binomials is crucial for polynomial multiplication because it allows you to apply strategic multiplication methods, like the distributive property, to simplify expressions efficiently.

Combining Like Terms

Combining like terms is an essential step in simplifying polynomial expressions. This involves merging terms that have the same variable and degree. In the expression \(4c^4 - \frac{8}{3}c^3 + 6c^2 - 48c^2 + 32c - 72\), some terms are "like terms."

Like terms share the same variables raised to the same power. Here’s what you do:

• Combine \(6c^2\) and \(-48c^2\) to get \(-42c^2\), since they both have \(c^2\).
• All other terms are already in their simplest form as they do not share the same variables raised to the same power with any other term in the expression.

By recognizing and combining like terms, you simplify the polynomial, making it easier to read and use.

Simplification

Simplification is the process of reducing an expression to its simplest form, where no further arithmetic operations can be conducted. After combining like terms, you streamline the expression, factoring in all the necessary operations.

The final result of our exercise comes out to \(4c^4 - \frac{8}{3}c^3 - 42c^2 + 32c - 72\), representing the expression in its simplest, most accessible form.

Simplification helps in understanding the structure of the polynomial and can make future operations, like solving or integration, more straightforward. Always aim for the simplest possible expression as a solution, as it's not only neat but also crucial in mathematical problem-solving.