Problem 9 - Elementary Algebra

Question

Factor the following, if possible. $$ 3 x^{2}+6 x y+2 y^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the expression is \((3x + 2y)(x + 2y)\).

Step 1: Identify common factors in the expression

In the given expression, we see that there is a common factor of 3 for the first two terms: $$ 3 x^{2}+6 x y+2 y^{2} = 3(x^2 + 2xy) + 2y^2 $$

Step 2: Rearrange the terms, if necessary

Since the expression is already in the required order (i.e., having a common binomial factor), we do not need to rearrange the terms. We can continue to factor by grouping.

Step 3: Factor by grouping

We can now factor out the common binomial factor from the two groups we identified in step 1: $$ 3(x^2 + 2xy) + 2y^2 = (3x)(x + 2y) + (2y)(x + 2y) $$

Step 4: Factor out the common binomial factor

Here, we see that the two terms have the common binomial factor \((x + 2y)\). We can factor this out: $$ (3x)(x + 2y) + (2y)(x + 2y) = (3x + 2y)(x + 2y) $$ So, the factored form of the given expression is: $$ 3 x^{2} + 6 x y + 2 y^{2} = (3x + 2y)(x + 2y) $$

Key Concepts

Binomial Factorization

Binomial factorization is a method used to simplify algebraic expressions by finding two binomials that, when multiplied together, restore the original expression. The process often involves recognizing patterns in the expressions and using algebraic identities. In our given problem, the expression is turned into two binomials, \( (3x + 2y) \) and \( (x + 2y) \). These multiply to return to the original expression \( 3x^2 + 6xy + 2y^2 \).

When tackling binomial factorization problems, it is essential to:

• Identify perfect squares or other recognizable patterns, like \( a^2 + 2ab + b^2 = (a+b)^2 \).
• Use associative and distributive properties to rearrange terms as needed.
• Be precise with calculations to ensure accuracy in factor identification.

Mastering these steps enhances problem-solving skills, making complex algebra more manageable.

Factoring by Grouping

Factoring by grouping is an effective technique for simplifying polynomial expressions that can be split into common groups. This method proves helpful when the expression doesn't have a single common factor across all terms but can be separated into groups that can be factored separately. In the exercise, the expression \( 3x^2 + 6xy + 2y^2 \) was divided into two groups: \( 3(x^2 + 2xy) \) and \( 2y^2 \).

To factor by grouping successfully:

• Group terms that show a common factor.
• Factor each group separately, revealing a common binomial factor where possible.
• Combine the factors, which, in this case, allowed us to recognize \( (x + 2y) \) as the common factor.

This approach gradually breaks down complex expressions into understandable components, simplifying the factorization process.

Common Factors

Identifying common factors is one of the foundational steps in algebraic factoring. A common factor is a term or expression that divides all terms of a polynomial without leaving a remainder. In our initial expression, \( 3x^2 + 6xy + 2y^2 \), the term \( 3 \) was a common factor for the first two terms, leading to an initial grouping of \( 3(x^2 + 2xy) \).

When finding common factors:

• Examine each term's constants and variables systematically.
• Factor out numbers or variables that appear in each part of the expression.
• Use common factors to simplify expressions before further operations like grouping or factorization.

Understanding how to identify and use common factors is critical for managing complex algebraic expressions and achieving correct results efficiently.