Problem 60 - Algebra 2

Question

Factor completely. $$ x^{2}+5 x $$

Step-by-Step Solution

Answer

The expression factors to \(x(x + 5)\).

Step 1: Identify Common Factors

Look for common factors in the terms of the expression. In the expression \(x^2 + 5x\), note that both terms, \(x^2\) and \(5x\), have an \(x\) as a common factor.

Step 2: Factor Out the Common Factor

Since \(x\) is common to both terms, factor \(x\) out of the expression. This gives \(x(x + 5)\).

Step 3: Verify the Factored Expression

Multiply the factors back together to ensure the factored form is correct. Calculate \(x(x + 5) = x^2 + 5x\), which confirms the factorization is correct.

Key Concepts

Common Factors

One of the simplest yet crucial concepts in algebra is identifying common factors in expressions. A common factor is an element that appears in each term of an algebraic expression. Recognizing these can greatly simplify the process of factoring polynomials.
For instance, in the expression \(x^2 + 5x\), the term \(x\) is present in both \(x^2\) and \(5x\). Hence, \(x\) is a common factor.
By factoring out \(x\) from each term, you are effectively grouping the terms under a shared component, which simplifies the expression into smaller, more manageable pieces to work with.
Identifying common factors is the first step in the process of factoring polynomials. Make it a habit to always look for shared numbers, variables, or both in the terms of the expression. This provides a strong foundation for solving more complex algebraic equations.

• Identify common factors in each term.
• Factor these elements out to simplify equations.
• Remember this technique aids in managing complicated polynomials.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations, such as addition and multiplication. These are foundational components of algebra that allow us to solve equations and understand relationships between numbers.
In expressions like \(x^2 + 5x\), the structure is key to how we approach solutions. \(x^2\) signifies a number \(x\) multiplied by itself, while \(5x\) implies multiplication of \(x\) by 5. These terms, when combined, create the algebraic expression.
Understanding how these expressions are formed and how to manipulate them is crucial. It allows us to transform complicated equations into simpler forms by applying various strategies, such as factoring or rearranging terms.
Often, algebraic expressions contain several terms that may share common factors, enabling you to reduce them neatly. This plays into both simplifying expressions and performing operations like addition, subtraction, or multiplication across algebraic equations.

• Identify variables and constants in expressions.
• Operate on expressions using mathematical operations.
• Practice manipulating expressions to simplify or solve equations.

Polynomial Factorization

Polynomial factorization is the process of breaking down an algebraic expression into simpler multiplicative components. It helps in solving equations and understanding the roots and properties of polynomial expressions.
The exercise \(x^2 + 5x\) is a perfect example of polynomial factorization. By using the common factor \(x\), you can simplify it to \(x(x + 5)\). Each part of this expression is a factor, and together they represent the original polynomial.
Factoring polynomials is not just about recognizing common factors but also about rearranging and rewriting expressions. It requires analyzing the terms to see how they can multiply back into the original polynomial, ensuring the factorization is correct.
Proper factorization makes it easier to find solutions to polynomial equations, such as determining the values that make the expression zero. This practice is essential in topics such as quadratic equations and algebraic fractions.

• Identify potential factors within each expression.
• Break down polynomials into these factors.
• Use factorization to simplify solving equations.