Problem 8 - College Algebra

Question

Factor out the greatest common factor. $$ x(2 x+1)+4(2 x+1) $$

Step-by-Step Solution

Answer

The fully factored expression is \( (2x+1)(x+4) \)

Step 1: Identify the common factor

Take a look at the problem and find the common factor. In our case, it is \( (2x+1) \) as it is present in both terms.

Step 2: Factor out the common factor

Factoring out means we express the expression as a product, where one factor is the common factor identified earlier. After factoring, the original expression of \( x(2 x+1)+4(2 x+1) \) simplifies to \( (2x+1)(x+4) \).

Step 3: Simplify the Expression

In this case, \( (2x+1)(x+4) \) is already the simplest form so there is no further need for simplifying. Thus, the expression is fully factored.

Key Concepts

Greatest Common Factor

When it comes to factoring polynomials, an important tool is the Greatest Common Factor (GCF). The GCF of a polynomial is the highest term that divides all terms of the polynomial without leaving a remainder. In simpler terms, it's the biggest thing you can "pull out" from each part of your algebraic expression.

To find the GCF, follow these steps:

• Identify the common terms: Look at each term in your expression and spot what's common among them.
• Choose the smallest power: If a variable appears in more than one term, use the smallest exponent appearing on that variable.
• Use coefficients: For numbers, simply use the largest number that divides all coefficients.

The GCF for the expression \( x(2x+1) + 4(2x+1) \) is \((2x+1)\) because it's a term that repeats in both parts. Factoring it out simplifies the expression by reducing complexity and making further computations easier.

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra that can represent numbers, operations, and variables. Each expression consists of terms, constants, coefficients, and variables, and can express various mathematical concepts or real-world scenarios.

Let's break down some elements of algebraic expressions:

• Variables: Symbols like \(x\), typically representing unknown numbers.
• Constants: Fixed numbers, like 4 in the expression \(4(2x+1)\).
• Coefficients:** Numbers multiplying variables, such as 2 in \(2x\).
• Terms: Each separate part, like \(x(2x+1)\) or \(4(2x+1)\).

Understanding these components helps in rearranging and simplifying expressions effectively. The expression \(x(2x+1) + 4(2x+1)\) is a classic example, where grouping similar terms through factoring can make complex problems manageable.

Polynomial Factorization

Polynomial factorization is a key technique in simplifying expressions and solving algebraic equations. It involves breaking down a polynomial into simpler factors that can be multiplied together to obtain the original expression. This technique uncovers important properties and roots of the polynomial, making it easier to handle.

Here's how polynomial factorization typically works:

• Identify the common factor: Look for common terms or binomials across the polynomial.
• Factor out the common elements: Express the polynomial as a product of its factors, like extracting \((2x+1)\) in the given example.
• Simplify: Ensure the resulting expression is in its simplest form, which sometimes involves further factoring or reduction.

In the expression \(x(2x+1) + 4(2x+1)\), the use of \((2x+1)\) as a common factor allows us to factorize the expression into \((2x+1)(x+4)\), simplifying it and revealing its structure. This step makes solving equations or further manipulating the expression more straightforward.