Problem 9 - Elementary Algebra

Question

List, if any appear, the common factors in the following expressions. $$ 4(a+1)^{3}+10(a+1) $$

Step-by-Step Solution

Answer

Answer: The common factors are 2 and $$(a+1)$$.

Step 1: Identify the terms in the expression

The given expression consists of two terms: 1. $$4(a+1)^3$$ 2. $$10(a+1)$$

Step 2: Examine terms for common factors

Look at the two terms to identify any common factors. We can see that both terms have a factor of 2 and a factor of $$(a+1)$$.

Step 3: Factor out the common factors

Next, factor out the common factors of 2 and $$(a+1)$$ from the expression: $$ 2(a+1)(2(a+1)^2 + 5) $$

Step 4: List the common factors

After factoring out the common factors, we can see that the common factors in the given expression are 2 and $$(a+1)$$.

Key Concepts

Factoring Expressions

When we talk about factoring expressions, we're referring to breaking down a complex expression into simpler ones. This is similar to finding the common building blocks of numbers when you look for the greatest common divisor. In algebra, factoring expressions simplifies how we handle equations. To factor an expression:

• Identify the individual terms. Look for terms that might share components or numbers that evenly divide all parts of the expression.
• Find what the terms have in common. It could be numbers or even algebraic terms, like variables or expressions within parentheses.
• Extract these common factors from each term, rewriting the entire expression in a new way that highlights these commonalities.

Factoring expressions is crucial for simplifying complex problems, making them easier to solve and understand. Simplifying often reveals the underlying structure of an equation, providing insight or making further calculation steps much simpler.

Algebraic Terms

Algebraic terms are the basic building blocks of mathematical expressions. An algebraic term can consist of constants, variables, or powers of variables. Sometimes, they include coefficients, which are numbers multiplying the variables. For example, in the expression \(4(a+1)^3 + 10(a+1)\), the first term is \(4(a+1)^3\) and the second term is \(10(a+1)\). Each term:

• Includes a numerical coefficient (like 4 and 10 here), which dictates the term's magnitude.
• Might contain variables or expressions within parentheses, like \((a+1)^3\) and \((a+1)\).

Understanding algebraic terms is key when factoring expressions. By breaking down each part of a term, you can identify similarities between different terms and factor them out effectively. Remember, the power of algebraic manipulation lies in recognizing these structures and properties.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operators (like \(+\), \(-\), \(*\), etc.) that represents a particular value or set of operations. Unlike an equation, it doesn't assert equality to anything but rather invites you to compute or simplify. Expressions can range from simple, like \(2x + 3\), to complex, like \((4(a+1)^3) + (10(a+1))\). A few characteristics include:

• Expressions can represent operations and are often part of larger equations.
• They can include nested components, similar to \((a+1)^3\), which itself is a smaller expression within a larger one.
• Simplification often involves identifying common factors or distributing terms appropriately.

In everyday mathematics, reducing expressions to their simplest forms is usually helpful. It makes further manipulation, comparison, or solving much easier, preparing the expression for effective utility in various scenarios.