Problem 121 - Introductory and Intermediate Algebra for College Students 4th

Question

Factor completely: $$6 x^{2}-8 x y+2 y^{2}$$ (Section 6.5, Example 8)

Step-by-Step Solution

Answer

The completely factored form of \(6x^{2}-8xy+2y^{2}\) is \(2(\sqrt{3}x-y)^{2}\)

Step 1: Identify common factors

The expression is \(6x^{2}-8xy+2y^{2}\). Inspect each term to see if there's a common factor, which in this case is 2. This results in \(2(3x^{2}-4xy+y^{2})\)

Step 2: Identify perfect square trinomial

Next we look at \(3x^{2}-4xy+y^{2}\) to determine if it's a perfect square trinomial. This is a perfect square trinomial if it's of the form \((a-b)^2=a^2-2ab+b^2\). Here, we can rewrite it as \((\sqrt{3}x-y)^{2}\)

Step 3: Factor completely

Finally, replace \((\sqrt{3}x-y)^2\) back in to complete the factorization. So, the factorized form of the expression is \(2(\sqrt{3}x-y)^{2}\)

Key Concepts

Perfect Square Trinomial

Understanding the concept of a perfect square trinomial is essential in algebra when dealing with quadratic expressions. A trinomial is a polynomial with three terms, and it becomes a perfect square when it can be expressed as the square of a binomial. Simply put, if you have a trinomial in the form of \(a^2 + 2ab + b^2\), it factors neatly into \((a + b)^2\). Similarly, a trinomial in the form \(a^2 - 2ab + b^2\) factors to \((a - b)^2\).

In the given exercise, the trinomial \(3x^2 - 4xy + y^2\) is identified as a perfect square because it matches the form of \(a^2 - 2ab + b^2\) where \(a = \sqrt{3}x\) and \(b = y\). Consequently, the trinomial can be factored as \((\sqrt{3}x - y)^2\), a crucial step to simplifying the expression and solving equations.

Common Factors in Algebra

The journey to simplify an algebraic expression often starts with finding common factors. A common factor is a term that divides evenly into each term of the expression. It's comparable to finding what ingredients a set of different dishes have in common. In algebra, extracting these factors is a way to simplify expressions and solve equations more easily.

For instance, in the expression \(6x^2 - 8xy + 2y^2\), the number 2 is a common factor for each term. By dividing the entire expression by 2, we get \(2(3x^2 - 4xy + y^2)\), which reveals a neater, more manageable expression. This step not only simplifies the problem but also sets the stage for further factoring, such as identifying a perfect square trinomial in this case.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that together represent a specific value or set of values. These expressions are the building blocks of algebra and are used to describe patterns, define relationships, and solve problems.

Take our example \(6x^2 - 8xy + 2y^2\), this is an algebraic expression with variables \(x\) and \(y\), exponents, coefficients, and subtractive operations. Grasping how to work with such expressions is key. This involves understanding their structure, performing operations like factoring, and being able to rewrite them in equivalent, but more useful, forms—as seen with the extraction of common factors and identification of perfect square trinomials. Adequate practice with these concepts bolsters problem-solving skills and deepens one's mathematical comprehension.