Problem 19 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ x^{2} y^{2}-10 x y+25 $$

Step-by-Step Solution

Answer

The trinomial factors to \((xy - 5)^2\).

Step 1: Recognize the Trinomial Structure

The given trinomial is \(x^2y^2 - 10xy + 25\). Notice that it is structured as a quadratic trinomial in terms of \(xy\). It can be seen in the form \((xy)^2 - 10(xy) + 25\).

Step 2: Identify the Perfect Square Form

The trinomial resembles the perfect square form \((a-b)^2 = a^2 - 2ab + b^2\). Here, \((xy)^2\) serves as \(a^2\), and \(25\) serves as \(b^2\).

Step 3: Calculate 'b' and Verify

Since \(25 = 5^2\), we assume \(b = 5\). Verify that the middle term \(-10xy\) can be expressed as \(-2ab\), which computes to \(-2(xy)(5) = -10xy\). This confirmation aligns the trinomial with the perfect square form.

Step 4: Factor the Trinomial as a Perfect Square

With confirmation from the previous steps, the trinomial \(x^2y^2 - 10xy + 25\) can be factored as a perfect square: \((xy - 5)^2\).

Key Concepts

Quadratic Trinomials

Quadratic trinomials are expressions that take the form of a three-term polynomial, typically written as \(ax^2 + bx + c\). In these expressions, the largest exponent is 2, which signifies the quadratic nature. Recognizing a quadratic trinomial is the first step toward factoring it.
To identify a quadratic trinomial, look for:

• The highest degree, which should be 2.
• Three terms which include a leading square term, a linear term, and a constant.
• Usually written in terms like \((xy)^2\) as we see in some factored forms.

In the exercise, the trinomial \(x^2y^2 - 10xy + 25\) is structured in a way that suits our understanding of quadratic trinomials, where \((xy)^2\) acts like the \(x^2\) term of a quadratic.

Perfect Square Trinomials

Perfect square trinomials are special types of quadratic trinomials that can be expressed as the square of a binomial. The standard form for a perfect square trinomial looks like \((a - b)^2 = a^2 - 2ab + b^2\). Recognizing this form can significantly simplify the factorization process.
For example:

• The first term of the trinomial is a squared term, such as \((xy)^2\).
• The third term is also a perfect square, like \(25\), which is \(5^2\).
• The middle term should equal \(-2ab\), matching the equation \(-10xy = -2(xy)(5)\).

In our exercise, \(x^2y^2 - 10xy + 25\) fits the perfect square form, allowing us to factor it as \((xy - 5)^2\). Recognizing the pattern lays the foundation for this conclusion.

Factorization Techniques

Transforming a trinomial into its factors involves applying various factorization techniques. Understanding these methods enhances skills in simplifying polynomial expressions and equations. One common method is factoring by recognizing patterns, such as perfect squares, as shown in the exercise above.
Common factorization techniques include:

• Recognizing Patterns: Identifying known patterns like perfect square trinomials helps in quickly simplifying an expression.
• Grouping: When conventional patterns don't work, grouping terms and factoring them separately can be efficient.
• Trial and Error: Sometimes, testing potential factor combinations to solve a problem might be necessary.

In the provided example, recognizing the trinomial as a perfect square reduced the complexity immensely. Always examine the structure of the trinomial first to choose the most effective technique.