Question

\(\left(2 x^{2}+4 x y+2 y^{2}\right) \times \frac{1}{2 x+2 y}=\) (A) \(y+x\) (B) \(\frac{2 x+4 y+2}{x+y}\) (C) \(2 x+4 x y+2\) (D) 2

Short answer

A: The result is \(x+y\).

Step-by-step solution

Simplify the Expression

Start by simplifying the expression in the numerator: \[2x^2 + 4xy + 2y^2\]Notice that this can be factored as:\[2(x^2 + 2xy + y^2)\] Realize this expression inside can be rewritten using the perfect square formula:\[(x+y)^2\] So, we have:\[2(x+y)^2\]

Apply the Multiplication

Multiply the factored form by \(\frac{1}{2x+2y}\):\[2(x+y)^2 \times \frac{1}{2x+2y}\]Notice that the denominator \(2x+2y\) can be factored out as:\[2(x+y)\]

Cancel Common Terms

Since we have \(2(x+y)^2\) in the numerator and \(2(x+y)\) in the denominator, cancel out the common terms:\[\frac{2(x+y)^2}{2(x+y)} = x+y\]Simplify the expression by removing both \(2\) and one \(x+y\) from the numerator and denominator.

Final Result

The simplification leads to the final expression:\[x + y\]This matches option (A) of the multiple-choice answers.

Key concepts

Polynomial Factoring

Polynomial factoring is breaking down a complicated polynomial into products of simpler polynomials. When we say a polynomial is factored completely, it means it cannot be divided any further using integer coefficients.

Factoring polynomials can simplify complex expressions and solve polynomial equations more easily. For example, in the exercise, you have the polynomial \(2x^2 + 4xy + 2y^2\). The first step in factoring is to look for any common factors, and here, each term can be divided by 2. It simplifies to \(2(x^2 + 2xy + y^2)\).

Next, observe the expression inside the parentheses, \(x^2 + 2xy + y^2\). This is a perfect square trinomial, which factors into \((x+y)^2\). Recognizing patterns, like perfect squares, can make factoring faster and clearer. Understanding these patterns allows for ease in manipulating expressions.

Rational Expressions

Rational expressions involve the division of two polynomials and often require simplification by factoring. Consider the expression given in the exercise: \(\frac{2(x+y)^2}{2(x+y)}\). Simplifying this involves canceling similar terms present in both the numerator and the denominator.

Here is how we do it:

• Identify the common factor: Look for expressions that are identical both above and below the division line.
• Cancel the common terms: Once identified, you can cancel them out because any number divided by itself is 1.

For our case, the term \(2(x+y)\) exists in both the numerator and the denominator. This results in the simplified form \(x+y\), showcasing how rational expressions can often be much simpler after cancellation.

Simplification Techniques

Simplification techniques in algebra are crucial for breaking down expressions to their simplest form. Essential strategies include factoring, canceling common terms, and recognizing algebraic identities. Let's dive into these techniques using our exercise as reference.

• Factor Common Terms: This technique involves finding and dividing out factors common to all parts of an expression. For instance, in \(2(x^2 + 2xy + y^2)\), the number 2 is a common factor, facilitating its extraction and simplification.
• Utilize Algebraic Identities: Recognizing expressions that match specific algebraically defined identities, like perfect square trinomials, allows us to simplify dramatically. In \((x^2 + 2xy + y^2)\), identifying it as \((x+y)^2\) helps in swift simplification.
• Reduce Fractions: In rational expressions, simplify by canceling terms common to the numerator and denominator, as seen in \(\frac{2(x+y)^2}{2(x+y)}\), which simplifies to \(x+y\) after canceling \(2(x+y)\).

By systematically applying these techniques, complex algebraic expressions become more manageable, yielding more elegant solutions and enhanced comprehension of the problem.