Problem 20 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The trinomial factors completely to \((2xy - 7)^2\).

Step 1: Identify the Trinomial Form

The given trinomial is \( 4x^2y^2 - 28xy + 49 \). This expression is in the standard quadratic form \( ax^2 + bx + c \), where \(a = 4x^2\), \(b = -28x\), and \(c = 49\).

Step 2: Check for a Perfect Square Trinomial

A perfect square trinomial is of the form \((ax + b)^2\). We suspect \( 4x^2y^2 - 28xy + 49 \) is a perfect square because it can potentially be written as \((2xy - 7)^2\).

Step 3: Expand the Perfect Square

To confirm, we expand \((2xy - 7)^2\) to see if it matches the given trinomial:\[(2xy - 7)(2xy - 7) = (2xy)^2 - 2\times(2xy)(7) + 7^2\]Calculating each term, we get:\[4x^2y^2 - 28xy + 49\]This indeed matches the original trinomial.

Step 4: Write the Factored Form

Since \( 4x^2y^2 - 28xy + 49 \) is a perfect square, we can write the factored form as:\[(2xy - 7)^2\]

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a special kind of algebraic expression that can be beautifully rewritten as the square of a binomial. Consider the trinomial in the exercise: \(4x^2y^2 - 28xy + 49\). This is actually a perfect square trinomial. A perfect square trinomial takes the form \((ax + b)^2\) or \((ax + b)(ax + b)\).

To identify it, we can look for patterns:

• The first term (\(4x^2y^2\)) is the square of \((2xy)\).
• The last term (\(49\)) is the square of \(7\).
• The middle term (\(-28xy\)) should be twice the product of the numbers inside the square terms. So, \(-28xy\) is twice \(2xy \cdot 7\).

This pattern confirms that this trinomial is a perfect square, allowing it to be factored as \((2xy - 7)^2\). Recognizing perfect square trinomials saves time and effort when factoring.

Factoring Techniques

Factoring is an essential skill in algebra that allows us to simplify expressions and solve equations. Different techniques can be applied depending on the form and structure of the algebraic expression you are working with.

When dealing with trinomials, these are some common techniques:

• **Factoring out a Greatest Common Factor (GCF):** Always start by looking for a common factor in all terms.
• **Factoring by Grouping:** Useful for expressions with four terms; pair and factor common elements.
• **Difference of Squares:** Works when the trinomial is structured as a difference and involves squares.
• **Perfect Square Trinomials:** Used when the trinomial is a perfect square, as we've identified in our example.

In our exercise, recognizing and applying the perfect square trinomial technique helped in directly finding \((2xy - 7)^2\). Always consider these techniques in context to efficiently solve factoring problems.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operations. It can be as simple as \(x + 3\) or as complex as our trinomial \(4x^2y^2 - 28xy + 49\). These expressions form the building blocks of algebra, allowing us to represent and analyze mathematical relationships.

Expressions can be simplified, expanded, and, importantly, factored to help solve equations or understand relationships better. They can contain:

• **Constants:** Fixed values, like \(49\).
• **Variables:** Symbols that can represent unknowns, like \(x\) and \(y\).
• **Coefficients:** Numbers that multiply the variables, like \(4\) in \(4x^2y^2\).

Understanding how to manipulate these components is key to mastering algebra. Factoring, a primary operation done on algebraic expressions, reveals deeper insights into their structure and roots.

Quadratic Form

A quadratic form is a specific type of polynomial expression where the highest degree of the variable is two. It frequently appears in the basic format \(ax^2 + bx + c\). Our expressed form \(4x^2y^2 - 28xy + 49\) is indeed a quadratic, despite involving two variables combined as a single term \((xy)\).

Key features of quadratic forms to examine include:

• **Leading Coefficient (\(a\)):** The coefficient of the squared term \((4x^2y^2)\).
• **Linear Coefficient (\(b\)):** The coefficient of the linear term \((-28xy)\).
• **Constant Term (\(c\)):** The standalone number, \(49\).

Quadratic forms can often be factored using methods such as completing the square, using the quadratic formula, or, as in our case, recognizing perfect square trinomials. These are integral in solving quadratic equations and finding roots or intersections.