Problem 67 - Elementary and Intermediate Algebra

Question

Factor. $$ 9 x^{2} y^{2}+30 x y+25 $$

Step-by-Step Solution

Answer

The expression factors to \((3xy + 5)^2\).

Step 1: Recognize the Form

Notice that the expression is a trinomial. It resembles the perfect square trinomial form which is of the type \((ax)^2 + 2abx + b^2\). Here, let's check if we can express the trinomial in this form.

Step 2: Identify Components

Compare the given trinomial \(9x^2y^2 + 30xy + 25\) with the perfect square form \((ax)^2 + 2abx + b^2\). Identify \(a = 3xy\) and \(b = 5\), since \((3xy)^2 = 9x^2y^2\) and \(5^2 = 25\).

Step 3: Check the Middle Term

Verify if the middle term equals \(2ab\). For \(a = 3xy\) and \(b = 5\), calculate \(2ab = 2 \times 3xy \times 5 = 30xy\). The middle term matches, confirming that it's a perfect square trinomial.

Step 4: Write the Factored Form

Since all conditions are met for a perfect square trinomial, we can write it in its factored form: \((3xy + 5)^2\). This uses the formula \((ax + b)^2 = (ax)^2 + 2abx + b^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial expression that results from squaring a binomial. It takes the form

• \((ax)^2 + 2abx + b^2\)

The identifying feature of a perfect square trinomial is its ability to be rewritten as the square of a binomial, specifically:

• \((ax + b)^2\)

Recognizing this structure involves identifying the following:

• \((ax)^2\) as the square of the first term
• \(b^2\) as the square of the second term
• \(2abx\) as the product of the first term and second, doubled

In the given exercise, we identified that

• \(a = 3xy\)
• \(b = 5\)
• middleware term \(2ab = 30xy\)

Thus, the expression \(9x^2y^2 + 30xy + 25\) was confirmed to be a perfect square trinomial because it perfectly fits the pattern.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They're the building blocks of algebra and are used to express relationships between quantities. A trinomial, like the one in our problem, is a type of algebraic expression with three terms. To better manage algebraic expressions, it's crucial to understand the following:

• Variables: Symbols like \(x\) and \(y\) represent quantities that can change or vary.
• Coefficients: The numerical factor that multiplies a variable, such as \(9\) in the term \(9x^2y^2\).
• Constants: Numbers on their own, such as \(25\) in the exercise.
• Terms: Parts of the expression separated by plus or minus signs.

Combining these elements in different configurations forms various types of algebraic expressions like monomials, binomials, and trinomials. Understanding their structure is key to factoring and simplifying them effectively.

Polynomial Factorization

Polynomial factorization is breaking down a polynomial into simpler components, or "factors," that, when multiplied together, give back the original polynomial. This process is essential for simplifying expressions and solving equations. In particular, perfect square trinomials offer an efficient method of factorization.
There are several methods for factorizing polynomials, but the perfect square method applies when the expression fits the pattern

• \((ax + b)^2\)

This allows us to rewrite the trinomial neatly and quickly as a squared binomial. The benefit is immediate, as it simplifies the task of solving equations set to zero. Furthermore, factorization can reveal roots or zeros of the polynomial, which are critical in graphing and understanding polynomial functions.
In the exercise, recognizing the trinomial as a perfect square made the factorization straightforward:

• \((3xy + 5)(3xy + 5)\)

Understanding polynomial factorization, especially with special cases like perfect squares, is crucial in algebra to make complex expressions more manageable and to solve related mathematical problems efficiently.