Problem 36 - College Algebra

Question

Factor each trinomial, or state that the trinomial is prime. $$ 3 x^{2}+4 x y+y^{2} $$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(3x^2 + 4xy + y^2\) is \((\sqrt{3}x + y)^2\).

Step 1: Recognize the trinomial

First step is to recognize what type of trinomial we have. In our case, the trinomial \(3x^2 + 4xy + y^2\) can be recognized as a perfect square trinomial.

Step 2: Rewrite as a square trinomial

Rewrite the trinomial as \(3x^2 + 2*2xy + (y^2)\). This look similar to the formula \((ax + by)^2 = a^2x^2 + 2abxy + b^2y^2\). We have to find 'a' and 'b' such that this formula gives our trinomial. In our case, 'a' can be taken as \(\sqrt{3}\)x and 'b' as y.

Step 3: Factorize the trinomial

Using formula and found 'a' and 'b', factorize the trinomial as \((\sqrt{3}x + y)^2\).

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are special kinds of polynomials that can be factored into a binomial squared. A trinomial is said to be a perfect square if it matches the specific pattern:

• \(a^2x^2 + 2abxy + b^2y^2\)

This pattern mirrors the squared form of a binomial \((ax + by)^2\). Recognizing this pattern is key to identifying whether a trinomial is a perfect square. Once identified, it can be rewritten in its factorized form effortlessly.
In the original exercise, the trinomial \(3x^2 + 4xy + y^2\) was first examined to see if it adheres to this formula structure. Upon inspection, it was noted that it could indeed fit the pattern by determining 'a' and 'b' so that \((ax + by)^2\) results in our given expression when expanded. Once the criteria for a perfect square trinomial are confirmed, factorization becomes straightforward.

Polynomial Expressions

Polynomial expressions are mathematical phrases derived by combining numbers, variables, and exponents. These expressions can take various forms, such as binomials (having two terms) or trinomials (having three terms), like the one in our exercise:

• \(3x^2 + 4xy + y^2\)

Different polynomials follow different rules and patterns, making understanding these properties essential for factoring. The most efficient factorizations exploit special forms, such as perfect square trinomials, as seen here. When manipulating polynomial expressions, always look for these unique patterns and consider utilizing algebraic identities. Additionally, practice with various polynomials will enhance the ability to spot these patterns and rework the expressions seamlessly into their factorized states.

Algebraic Factorization

Algebraic factorization is the process of breaking down a complex polynomial into simpler components—often a product of simpler binomials. This concept is fundamental for simplifying expressions and solving equations efficiently. To factor, one might first look for patterns and utilize identities such as:

• The difference of squares:
• Perfect square trinomials:

These identities serve as shortcuts to achieving a more manageable form of polynomials.
This particular exercise provided a clear example of algebraic factorization by transforming the trinomial \(3x^2 + 4xy + y^2\) into a squared binomial \((\sqrt{3}x + y)^2\). By choosing 'a' and 'b' carefully, factoring becomes a straightforward task, making the large polynomial expression much simpler to interpret and utilize in further calculations.