Problem 81 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$(3 x+1)^{6}\left(\frac{1}{2}\right)(2 x-5)^{-12}(2)+(2 x-5)^{1 / 2}(6)(3 x+1)^{5}(3)$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(3x+1)^6}{(2x-5)^{12}} + 18(2x-5)^{1/2} \cdot (3x+1)^5 \).

Step 1: Identify and Separate Terms

The given expression consists of two main additive components, which can be rewritten for simplicity: \\( (3x+1)^6 \cdot \frac{1}{2} \cdot (2x-5)^{-12} \cdot 2 \) and \\((2x-5)^{1/2}\cdot 6 \cdot (3x+1)^5\cdot 3\) \This will help in systematically approaching the simplification by dealing with each part separately.

Step 2: Simplify the First Term

Simplify the first part: \\( (3x+1)^6 \cdot \frac{1}{2} \cdot (2x-5)^{-12} \cdot 2 \). \This can be simplified as \\( (3x+1)^6 \cdot 1 \cdot (2x-5)^{-12} = \frac{(3x+1)^6}{(2x-5)^{12}} \). \The factors \(\frac{1}{2}\) and \(2\) cancel each other out, simplifying the expression.

Step 3: Simplify the Second Term

For the second term: \\( (2x-5)^{1/2} \cdot 6 \cdot (3x+1)^5 \cdot 3 \), \first simplify by multiplying the constants: \\( 6 \cdot 3 = 18 \). \Thus, the expression is \\( 18 \cdot (2x-5)^{1/2} \cdot (3x+1)^5 \).

Step 4: Combine and Simplify the Expression

Combine the expressions found in Steps 2 and 3: \\( \frac{(3x+1)^6}{(2x-5)^{12}} + 18(2x-5)^{1/2} \cdot (3x+1)^5 \). \Since these two terms do not easily combine via basic simplification due to differing bases and powers, this is the simplified form of the expression.

Key Concepts

Algebraic Expressions

Algebraic expressions are essential components in math, acting as a way to communicate mathematical ideas using symbols and numbers. They often involve variables, like \( x \) and \( y \), combined with constants and operation symbols. Consider the expression given in this problem: \( (3x+1)^6 \cdot \frac{1}{2} \cdot (2x-5)^{-12} \cdot 2 + (2x-5)^{1/2} \cdot 6 \cdot (3x+1)^5 \cdot 3 \). This consists of terms that include exponents, coefficients, and variables combined in various ways.

To simplify an algebraic expression, we need to understand how to handle each part independently before combining the results. Simplifying these expressions often involves factoring, expanding, or using special rules for operations. Always be sure to break down complex expressions into simpler components to find the solution.

Additive components, like the two terms in our expression, signify that the whole expression is a sum. Each term, when isolated, can be simplified individually before bringing them back together. This method allows us to handle large and complex expressions more effectively.

Exponents and Powers

Exponents and powers are shorthand ways of expressing repeated multiplication. In this exercise, terms such as \( (3x+1)^6 \) and \( (2x-5)^{-12} \) illustrate how exponents are used to denote that a term is multiplied by itself a certain number of times.

Exponential expressions come with specific rules:

• Product of Powers: When multiplying two expressions with the same base, add their exponents.
• Power of a Power: Raise a power to another power by multiplying the exponents.
• Negative Exponents: Convert to a fraction with a positive exponent in the denominator.

Thus, when we simplify \( (3x+1)^6 \cdot \frac{1}{2} \cdot (2x-5)^{-12} \), we apply the negative exponent rule to move \( (2x-5) \) to the denominator with a positive exponent: \( \frac{(3x+1)^6}{(2x-5)^{12}} \). Understanding these rules is key to simplifying expressions with exponents.

Polynomial Functions

Polynomial functions are expressions constructed with variables raised to natural number exponents and coefficients. They can be seen in parts of our expression, such as \( (3x+1)^6 \) and \( (3x+1)^5 \), where binomials are raised to a power.

Polynomials are classified by their degree, which is the highest power of the variable. For example, a polynomial of degree 6 means the highest power of its terms is \( x^6 \).

Each term in a polynomial is called a 'monomial', and polynomials can be added or subtracted like regular numbers. In our task, different polynomial bases, like \( 3x+1 \) and \( 2x-5 \), are present. This requires us to simplify each polynomial part individually when dealing with their exponents and coefficients.

Understanding polynomial behavior is crucial for manipulating algebraic expressions and provides a foundation for solving equations that involve these types of functions.