Problem 120 - Elementary and Intermediate Algebra

Question

How are the procedures used to simplify \(3 x+4 x\) and \(3 \sqrt{x}+4 \sqrt{x}\) similar?

Step-by-Step Solution

Answer

Both procedures involve factoring out a common term and simplifying the coefficient.

Step 1: Observe Common Terms

For both expressions, identify the common terms found within each component. In the expression \(3x + 4x\), both terms include the variable \(x\). Likewise, in the expression \(3\sqrt{x} + 4\sqrt{x}\), both terms include \(\sqrt{x}\) as the common factor.

Step 2: Factor Out the Common Term

In the expression \(3x + 4x\), factor out the common variable \(x\). You will get \((3 + 4)x\). Similarly, in the expression \(3\sqrt{x} + 4\sqrt{x}\), factor out the common term \(\sqrt{x}\) to obtain \((3 + 4)\sqrt{x}\).

Step 3: Simplify the Sum Inside Parentheses

Calculate the arithmetic inside the parentheses for both expressions. Compute \(3 + 4 = 7\). Therefore, both expressions simplify to \(7x\) and \(7\sqrt{x}\) respectively.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental process in algebra that means breaking down complex expressions into their simplest form. This is often done by combining **like terms** or **factoring** out common factors.

To simplify an expression, you first identify terms within the expression that can be simplified, such as like terms (terms that have the same variables raised to the same power). Once these terms are recognized, you can perform operations to combine or reduce them into fewer terms.

• For instance, in the algebraic expression \(3x + 4x\), the terms are like terms because both have the variable \(x\).
• You can add the coefficients (numerical factors) of these terms together to simplify the expression to \(7x\).

This process helps to make expressions easier to work with, especially when you need to perform further algebraic manipulations such as solving equations.

Factoring

Factoring plays a pivotal role in simplifying algebraic expressions and equations. It involves breaking down an expression into products of simpler or more fundamental expressions. This is akin to "unwrapping" the expression into factors that when multiplied together give the original expression.

In the simplification of expressions like \(3x + 4x\) and \(3\sqrt{x} + 4\sqrt{x}\), one common method involves **factoring out** the greatest common factor (GCF).

• You look for the GCF of the terms, which is \(x\) for the first expression and \(\sqrt{x}\) for the second.
• By factoring this out, you rewrite \(3x + 4x\) as \((3 + 4)x\) and \(3\sqrt{x} + 4\sqrt{x}\) as \((3 + 4)\sqrt{x}\).

Whether dealing with numbers, variables, or combinations, factoring is a key technique that greatly simplifies expressions and is widely used in solving equations.

Like Terms

Identifying and combining **like terms** is critical in the simplification process for algebraic expressions. Like terms are those that have identical variables raised to the same power, even though they might have different coefficients.

When you encounter like terms, adding or subtracting them is straightforward and can lead to a much simpler expression.

• Consider \(3x + 4x\); both terms are like terms because they contain the variable \(x\).
• Similarly, in \(3\sqrt{x} + 4\sqrt{x}\), both terms are like terms sharing \(\sqrt{x}\).

By adding the coefficients of these terms, you simplify them down to fewer terms: \(7x\) and \(7\sqrt{x}\), respectively. This not only reduces complexity but also makes further mathematics operations easier to perform.