Problem 37 - Elementary Algebra

Question

For the following problems, list, if any should appear, the common factors in the expressions. $$ x^{2}+5 x^{2}-2 x^{2} $$

Step-by-Step Solution

Answer

Answer: The common factor is \(x^2\).

Step 1: Rewrite the Expression

First, let's rewrite the expression to simplify it: $$ x^2 + 5x^2 - 2x^2 $$ can be rewritten as: $$ (x^2) + (5x^2) + (-2x^2) $$

Step 2: Identify the Common Factors

Now let's look for the factors that are common in all of the terms present in the expression. Each term has a factor of \(x^2\).

Step 3: List the Common Factor

The only common factor in all the terms of the expression is \(x^2\).

Key Concepts

Factoring Expressions

Factoring expressions is a vital concept in algebra. It involves breaking down an expression into a product of simpler components, called factors. For instance, given a polynomial like \(x^2 + 5x^2 - 2x^2\), factoring means identifying what common factors exist in each term.

To factor expressions efficiently:

• Look for the greatest common factor (GCF) across all terms. In our example, each term has the factor \(x^2\).
• Use this common factor to simplify or further transform the expression, which helps in solving algebraic equations.

Factoring is not only about reducing or simplifying expressions. It's a foundation for more advanced topics such as solving quadratic equations and simplifying derivatives in calculus.

Algebraic Expressions

To understand factoring, it's crucial to first grasp what algebraic expressions are. These expressions consist of numbers, variables, and operators (like addition and subtraction). For example, \(x^2 + 5x^2 - 2x^2\) is an algebraic expression because it includes a variable (\(x\)) raised to a power and combined using arithmetic operations.

Algebraic expressions allow us to represent mathematical relationships in a generalized form:

• They can represent real-world scenarios, helping describe patterns or solve problems involving unknown values.
• Expressions can often be manipulated through factoring, expanding, or simplifying, leading to equivalent yet often more useful forms.

Having a strong understanding of how to work with algebraic expressions is fundamental for success in mathematics, as it provides a toolkit for exploring and resolving complex equations.

Polynomial Simplification

Simplifying polynomials is a technique used to make complex expressions easier to work with. This involves combining like terms or reducing expressions to their simplest form by way of factoring. Simplification can help reduce errors in calculations and make the solving process more straightforward.

For simplifying polynomials like \(x^2 + 5x^2 - 2x^2\):

• Identify and combine like terms. Here, all terms are multiples of \(x^2\). So, we combine them to simplify the polynomial to \(4x^2\).
• Use polynomial simplification to reveal the structure of an expression, assisting in further manipulation or factoring.

Simplifying polynomials is an invaluable step before moving on to solve equations or answer more complex mathematical queries. It helps clarify the mathematical relationships within the expression, making both analysis and computation easier.