Problem 9 - Algebra and Trigonometry

Question

\(5-12\) . Factor out the common factor. $$ 2 x^{2} y-6 x y^{2}+3 x y $$

Step-by-Step Solution

Answer

The expression is factored as \(xy(2x - 6y + 3)\).

Step 1: Identify Common Factor

Look at the terms in the expression: \(2x^2y\), \(-6xy^2\), and \(3xy\). Identify any common factors in their coefficients and variables.

Step 2: Extract the Common Factor

From the terms, we notice that each term has \(xy\) in common. Extract \(xy\) as a common factor from the expression. This gives us \(xy(2x - 6y + 3)\).

Step 3: Recheck Factoring

Multiply the factored terms back to ensure correctness: \(xy(2x - 6y + 3) = 2x^2y - 6xy^2 + 3xy\). The factorization is correct.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (such as +, −, ×, ÷). They don't have an equality sign, which distinguishes them from equations. In the given exercise, the expression is \[2x^2y - 6xy^2 + 3xy.\] This expression is made up of three separate terms, each containing a combination of the variables \(x\) and \(y\) along with their coefficients. To manage such expressions effectively, we utilize operations like addition, subtraction, and techniques like factoring.

The primary aim in algebraic expressions during factoring is to simplify them by identifying a common structure or factor among terms. This makes them easier to understand and solve when used in equations or inequalities. Each part of an algebraic expression can often be represented in a simpler form by recognizing common elements.

Common Factors

A common factor is a number or expression that divides each term in an expression without leaving a remainder. In this problem, we are tasked with identifying the common factors in the terms \[2x^2y, -6xy^2,\text{ and } 3xy.\]This involves finding the greatest factor that is present in each term.

We start by looking at the coefficients (2, -6, and 3) and the variables (\(x^2y, xy^2,\text{ and } xy\)). The number 1 is a common factor of all, but is not useful for simplification. Next, we notice that \(xy\) is common among all terms. This extraction simplifies the expression to \[xy(2x - 6y + 3).\]

Factoring out the common factor like this is crucial because it reduces the expression effectively while maintaining all its original properties. This is a key tactic in algebra to simplify complex expressions and solve broader problems.

Polynomials

A polynomial is an algebraic expression consisting of variables, coefficients and non-negative integer exponents. In our original expression\[2x^2y - 6xy^2 + 3xy,\] we have a polynomial with three terms or monomials. Each of these terms includes a variable component (with exponents) and a numeric coefficient.

Polynomials are categorized by the number of terms; for example, our expression is known as a trinomial because it has three terms. Degeneration among these terms varies, governed by the highest exponent present.

When working with polynomials, understanding the degree and structure helps in various operations, such as factorizing, which simplifies calculations and potential solving of equations. This makes polynomials foundational in algebra, handling everything from simple expressions to complex functions. Recognizing patterns within polynomials allows for targeted strategies like factoring to make them manageable.