Problem 102 - College Algebra

Question

Factor and simplify each algebraic expression. $$ -8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1} $$

Step-by-Step Solution

Answer

Therefore, the simplified form of the expression is \(72x^2 - 132x - 62\).

Step 1: Identification

Identify each term. The given expression can be seen as two terms: 1. -8 times \( (4x + 3)^{-2} \)2. 10 times \( (5x + 1) \) times \( (4x + 3)^{-1} \)

Step 2: Handling First Term

For the first term, apply the number outside the bracket to each term within the bracket. This can be done due to the Property of Exponents stating that \((ab)^n = a^n b^n\). Thus, the first term simplifies to \(-8 \times (16x^2 + 24x + 9)\). Simplifying further, the first term becomes \(-128x^2 - 192x - 72\).

Step 3: Handling Second Term

For the second term, follow the same process as in step 2 of multiplying the number outside the bracket to each term within the bracket. This simplifies the second term to \(10 \times (20x^2 + 2x + 4x + 1)\). Simplifying further, the second term becomes \(200x^2 + 60x + 10\).

Step 4: Combine and Simplify

Combine the simplifications from Step 2 and Step 3 by addition to give the final expression. This results in \(-128x^2 - 192x - 72 + 200x^2 + 60x + 10\), which further simplifies to \(72x^2 - 132x - 62\).

Key Concepts

Exponents

Exponents are a fundamental aspect of algebra and mathematics in general. They act like a shorthand for expressing repeated multiplication. For example, when you have a term like \(a^2\), it means \(a\) is multiplied by itself once. Similarly, \(a^3\) means \(a \times a \times a\). When you see \((4x+3)^{-2}\), it means \((4x+3)\) is in the denominator of a fraction and multiplied by itself twice. This negative exponent indicates a reciprocal. Remembering these basic rules helps in simplifying expressions quickly.

In the original exercise, exponents play a crucial role in the simplification process. We simplified each term by using properties of exponents, especially when applying negative exponents. For instance, using the rule that \(ab^n = a^n b^n\), we were able to tackle the complexity of terms like \((4x+3)^{-2}\). Handling exponents properly helps break down complex expressions into simpler forms.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations combined into meaningful mathematical statements. They can include terms with coefficients (numbers in front), variables (like \(x\)), and exponents.

An expression like \(-8(4x+3)^{-2}+10(5x+1)(4x+3)^{-1}\) is a combination of multiple algebraic terms, each with different coefficients and exponents. It’s crucial to identify each term separately before proceeding to manipulation. In the given problem, each term was isolated first to clearly understand its structure. Separating them allowed us to apply specific rules about multiplication and addition effectively. Hence, recognition and understanding of algebraic expressions are essential in simplifying complex mathematical problems.

Simplification

Simplification is the process of reducing expressions into their most concise form while maintaining their original value. It's like cleaning up a messy room, so it's easier to navigate.

In mathematics, simplification involves combining like terms, factoring, and following specific algebraic rules to reduce expressions to their simplest forms. In our exercise, we took two terms and applied proper mathematical operations to simplify them individually before combining them for the final simplification.

• First, we expanded each term using the distributive property.
• Next, we simplified by performing addition between common coefficients and similar terms.
• Finally, combining and further reducing the expression gave us the simplest form possible.

This process ensures the expressions are not only accurate but also easier to work with in further calculations or problem-solving.