Problem 43 - Algebra for College Students

Question

Factor completely. $$3 x(2 a+b)-2 y(2 a+b)$$

Step-by-Step Solution

Answer

The expression factors to \((2a+b)(3x-2y)\).

Step 1: Identify the Common Factor

Look at the expression: \(3x(2a+b) - 2y(2a+b)\). Notice that \((2a+b)\) is a common factor in both terms.

Step 2: Factor Out the Common Factor

Take the common factor \((2a+b)\) out of the expression. This gives us \((2a+b)(3x - 2y)\).

Step 3: Verify the Factorization

To ensure correctness, distribute \((2a+b)\) back into each term: \((2a+b)(3x)\) gives \(3x(2a+b)\) and \((2a+b)(-2y)\) gives \(-2y(2a+b)\), reconstructing the original expression.

Key Concepts

Common Factor

In mathematics, particularly in the study of expressions, a **common factor** is an element that appears in all terms within an expression. Identifying the common factor is a crucial step in simplifying and solving problems. For example, in the expression \(3x(2a+b) - 2y(2a+b)\), the term \((2a+b)\) appears in both parts of the expression.

• To find a common factor, check each part of the expression for similarities.
• Factors can be numbers, variables, or entire expressions.
• Once identified, you can "factor" them out, simplifying the expression.

Finding a common factor makes expressions much easier to handle and solve. Always search for common elements as your first step when dealing with complex algebraic problems.

Polynomial Factorization

**Polynomial Factorization** is a method where we express a polynomial as a set of multiplication factors. It's like taking a number apart to find the prime numbers that multiply back to the original.In the expression \(3x(2a+b) - 2y(2a+b)\), once we identify \((2a+b)\) as a common factor, we can apply polynomial factorization to represent the expression as \[(2a+b)(3x - 2y)\].

• This method simplifies polynomials by breaking them into basic components.
• It helps solve equations and simplify expression computations.
• Factored forms are crucial for solving polynomial equations, especially in higher algebra.

Understanding how to factor polynomials improves problem-solving efficiency and assists in identifying roots of equations.

Algebraic Expressions

An **Algebraic Expression** consists of numbers, variables, and operations (like addition or multiplication). These expressions represent equations, allowing us to model real-world problems.For instance, in \(3x(2a+b) - 2y(2a+b)\), the expression is composed of:

• Numerical coefficients: \(3\) and \(-2\).
• Variables: \(x\), \(y\), \(a\), and \(b\).
• Operations: Multiplication and subtraction.

The goal when working with algebraic expressions is often to simplify them or to solve equations. Techniques such as identifying common factors and polynomial factorization are key methods in working with these expressions.Simplifying an expression makes it easier to understand and use for further operations or problem-solving. It is the foundation for advanced algebraic manipulations and crucial for dealing with equations in both pure and applied mathematics.