Problem 36 - A Graphical Approach to College Algebra

Question

Factor each perfect square trinomial completely. $$4 x^{2} y^{2}+28 x y+49$$

Step-by-Step Solution

Answer

The factored form is \((2xy + 7)^2\).

Step 1: Identify the structure of a perfect square trinomial

A perfect square trinomial takes the form \(a^2 \pm 2ab + b^2\), which factors to \((a \pm b)^2\). Here, \(4x^2y^2 + 28xy + 49\) can potentially be expressed in this way.

Step 2: Find the squares

Identify the squares in the trinomial: \(4x^2y^2 = (2xy)^2\) and \(49 = 7^2\). Now we have \((2xy)^2 + 28xy + 7^2\).

Step 3: Check the middle term

The middle term in a perfect square trinomial is \(2ab\). Check if \(28xy\) fits: we have \(2(2xy)(7) = 28xy\). Thus, the middle term matches \(2ab\).

Step 4: Express as a perfect square

Since the structure matches \((a+b)^2\) for \(a=2xy\) and \(b=7\), the trinomial can be written as \((2xy + 7)^2\).

Step 5: Write the factored form

The expression \(4x^2y^2 + 28xy + 49\) can be expressed as \((2xy + 7)^2\), which is its completely factored form.

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the building blocks of an expression. A polynomial is an expression made up of variables and constants with operations like addition and multiplication. Factoring involves breaking it down into simpler parts or products that, when multiplied together, give the original expression. This is important for solving equations or simplifying expressions.

• Factors are like the ingredients that make up the whole polynomial.
• The goal is to rewrite the polynomial as a product of its factors.

For example, when we factor the perfect square trinomial \(4x^2y^2 + 28xy + 49\), we express it in its simplest product form. The polynomial here breaks down into \((2xy + 7)^2\). Recognizing the structure of expressions helps in determining the factors more easily.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators. They can be simple, like \(x + 2\), or more complex like our trinomial \(4x^2y^2 + 28xy + 49\). Working with algebraic expressions involves understanding how to manipulate these terms to simplify or solve problems.

• Variables represent unknown values and function as placeholders until they are solved.
• Constants are standalone numbers without variables, like 49 here.

In our example, each term in the expression \(4x^2y^2 + 28xy + 49\) plays a specific role in the structure of the expression. Understanding the organization of the terms helps us find connections, like recognizing a perfect square trinomial, which is key to factoring and simplifying.

Quadratic Expressions

Quadratic expressions are a type of polynomial with the highest exponent of the variable being 2. A standard form of a quadratic is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratics appear frequently in algebra due to their simple yet rich structure which allows for a variety of solving techniques.

• The squared term is what makes it quadratic, giving it its characteristic parabolic graph shape.
• Quadratics are often solved by factoring, completing the square, or using the quadratic formula.

The trinomial in this exercise, \(4x^2y^2 + 28xy + 49\), can be viewed as a quadratic in terms of \((xy)\), with the expression becoming \((2xy + 7)^2\) when factored. Recognizing the quadratic form allows us to apply specific techniques suitable for solving or factoring these expressions. Understanding how to work with quadratic expressions is essential for mastering algebraic problems.