Problem 55
Question
Factor the expression completely. $$ 4 x^{2}+4 x y+y^{2} $$
Step-by-Step Solution
Verified Answer
The expression factors completely to \((2x + y)^2\).
1Step 1: Identify the Expression Structure
We need to factor the expression \(4x^2+4xy+y^2\). Notice that it resembles the structure of a perfect square trinomial, which has the form \((a+b)^2 = a^2 + 2ab + b^2\). We will check to see if this expression can be rewritten as a perfect square trinomial.
2Step 2: Determine Possible Values of a and b
In the expression \(4x^2+4xy+y^2\), we compare it with \((a+b)^2 = a^2 + 2ab + b^2\). This gives us \(a^2 = 4x^2\) and \(b^2 = y^2\). So, \(a = 2x\) and \(b = y\). Check if the middle term \(2ab\) equals \(4xy\): \(2(2x)(y) = 4xy\). This confirms that our choices for \(a\) and \(b\) are correct.
3Step 3: Write the Expression as a Perfect Square
We can now express \(4x^2 + 4xy + y^2\) as a perfect square, using the values of \(a\) and \(b\) determined earlier. Thus, it can be rewritten as \((2x + y)^2\).
4Step 4: Verify the Factored Form
Expand \((2x + y)^2\) to double-check our factorization: \((2x + y)(2x + y) = 4x^2 + 2xy + 2xy + y^2 = 4x^2 + 4xy + y^2\). The original expression is confirmed, so the factorization is \((2x + y)^2\).
Key Concepts
Perfect Square TrinomialFactoring ExpressionsAlgebraic Structures
Perfect Square Trinomial
Understanding a perfect square trinomial is crucial for solving certain algebra problems. A perfect square trinomial is an expression that can be written in the form \((a+b)^2 = a^2 + 2ab + b^2\). This pattern occurs when a binomial is squared. Recognizing this specific structure in an algebraic expression allows us to rewrite it more simply and concisely.
In the given problem, we have the expression \(4x^2+4xy+y^2\). By identifying it as a perfect square trinomial, we see that it matches the structure \((a+b)^2 = a^2 + 2ab + b^2\). By finding \(a\) and \(b\) such that \(a^2 = 4x^2\) and \(b^2 = y^2\), we discover \(a = 2x\) and \(b = y\). The middle term, \(2ab\), confirms this factorization because it equals \(4xy\).
This clear matching means our expression can be rewritten as the square of the binomial \((2x + y)^2\). This particular form helps in simplifying, solving, and understanding the algebraic structure of expressions.
In the given problem, we have the expression \(4x^2+4xy+y^2\). By identifying it as a perfect square trinomial, we see that it matches the structure \((a+b)^2 = a^2 + 2ab + b^2\). By finding \(a\) and \(b\) such that \(a^2 = 4x^2\) and \(b^2 = y^2\), we discover \(a = 2x\) and \(b = y\). The middle term, \(2ab\), confirms this factorization because it equals \(4xy\).
This clear matching means our expression can be rewritten as the square of the binomial \((2x + y)^2\). This particular form helps in simplifying, solving, and understanding the algebraic structure of expressions.
Factoring Expressions
Factoring expressions is an essential technique in algebra that involves breaking down an expression into simpler terms or factors. This process is helpful in solving quadratic equations and simplifying algebraic expressions. When factoring, the goal is to express a polynomial as a product of simpler polynomials.
In the context of the exercise, the expression \(4x^2 + 4xy + y^2\) is factored into the binomial \((2x + y)^2\). By identifying it as a perfect square trinomial, we make our task straightforward because it allows us to express the polynomial as a squared binomial. This means rather than handling three separate terms, we deal with one squared binomial.
In the context of the exercise, the expression \(4x^2 + 4xy + y^2\) is factored into the binomial \((2x + y)^2\). By identifying it as a perfect square trinomial, we make our task straightforward because it allows us to express the polynomial as a squared binomial. This means rather than handling three separate terms, we deal with one squared binomial.
- It simplifies computations.
- It reveals further insights into the roots of the polynomial.
- It aids in solving equations where the expression might appear.
Algebraic Structures
Understanding algebraic structures is like discovering the hidden framework of mathematics. They provide a set of rules to operate within that allows for consistent manipulation of mathematical expressions. In algebra, recognizing patterns such as a perfect square trinomial is part of understanding these structures.
An algebraic structure can be seen as a family of operations and elements that abide by specific rules. For instance, knowing that certain trinomials are squares of binomials gives us a shortcut or method to simplify complex problems. This comprehension is essential in both basic and advanced algebra as it extends to topics such as quadratic equations, polynomial identities, and more.
An algebraic structure can be seen as a family of operations and elements that abide by specific rules. For instance, knowing that certain trinomials are squares of binomials gives us a shortcut or method to simplify complex problems. This comprehension is essential in both basic and advanced algebra as it extends to topics such as quadratic equations, polynomial identities, and more.
- Improve problem-solving strategies.
- Develop insights into more complex algebraic manipulations.
- Lay foundational knowledge for further studies in mathematics and sciences.
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Problem 55
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