Problem 46 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 4 x^{2}+4 x y+y^{2} $$

Step-by-Step Solution

Answer

The factorized form is \((2x + y)^2\).

Step 1: Recognize the Quadratic Form

The given expression is in the form \(4x^2 + 4xy + y^2\). This resembles a perfect square trinomial of the form \((a + b)^2\). We will identify such terms in the expression.

Step 2: Identify the Square Terms and Cross Product

First, identify the square terms: \(4x^2 = (2x)^2\) and \(y^2 = (y)^2\). The middle term \(4xy\) is twice the product of these bases: \(2xy = 2(2x)(y)\). This confirms the expression is a perfect square.

Step 3: Write the Expression as a Perfect Square

Since \(4x^2 + 4xy + y^2\) matches the identity for a perfect square \((a + b)^2\), we write it in its factorized form: \((2x + y)^2\). This step transforms the expression from expanded form to its factored form.

Step 4: Confirm the Expansion

There's a quick confirmation step to expand \((2x + y)^2\) back to \(4x^2 + 4xy + y^2\) to ensure no mistakes were made in the factorization process. Each step checks out correctly.

Key Concepts

Quadratic Form

A quadratic form is an algebraic expression that takes the shape of a polynomial, specifically in the degree of two. Typically, a quadratic form is written as \(ax^2 + bxy + cy^2\), where \(a\), \(b\), and \(c\) are coefficients.
In the original exercise, we have the expression \(4x^2 + 4xy + y^2\). Identifying this expression as a quadratic form helps you see it as a standard pattern that you can factor. Recognizing such patterns is key in solving equations where the unknown is quadratic in nature.
Quadratic expressions can often be simplified further using various factorization techniques, making them easier to solve or evaluate within algebraic equations.

Perfect Square Trinomials

Perfect square trinomials are special kinds of quadratic expressions. They appear as a square of a binomial: \((a + b)^2\) or \((a - b)^2\). When expanded, these trinomials take the form \(a^2 + 2ab + b^2\).
In the case of \(4x^2 + 4xy + y^2\), it fits the pattern of \((a + b)^2\), specifically \((2x + y)^2\).
Recognizing a perfect square trinomial in your algebraic work allows you to quickly factor the expression. This saves time and reduces the risk of mistakes. A perfect square trinomial not only simplifies calculations but ensures accurate results when solving quadratic equations.

Factorization Techniques

Factorization techniques are mathematical strategies used to break down polynomials into simpler components called factors. Understanding these techniques is crucial in algebra as it allows for greater simplicity in solving various equations.
For instance, identifying \(4x^2 + 4xy + y^2\) as a perfect square trinomial, you apply the technique of factoring it into \((2x + y)^2\).
Here are some common factorization techniques you should be familiar with:

• Common Factor: Identifying a term that appears in all parts of the polynomial.
• Difference of Squares: Two squared terms separated by a minus sign.
• Grouping: Factoring by grouping parts of the polynomial to find common factors.

Using these techniques can greatly aid in simplifying complex algebraic expressions, making them much easier to solve or interpret.