Problem 61 - Elementary Algebra

Question

Simplify. $$ 2(x+y) 2+3(x+y) 2 $$

Step-by-Step Solution

Answer

The simplified expression is \(5(x+y)^2\).

Step 1: Distribute each term

For both expressions \(2(x+y)^2\) and \(3(x+y)^2\), distribute the constants by applying them to each term within the parentheses. We will treat \((x+y)^2\) as a common factor first. This will give us: \(2(x+y)^2 + 3(x+y)^2\).

Step 2: Factor out the common term

Since \((x+y)^2\) is common in both terms, factor it out. Write the simplified expression as: \((2+3)(x+y)^2\).

Step 3: Simplify the coefficients

Add the coefficients 2 and 3 to obtain 5. This step simplifies the expression to \(5(x+y)^2\).

Step 4: Write the final simplified expression

Combine all parts to express the simplified answer: \(5(x+y)^2\).

Key Concepts

Factorization

Factorization is the process of breaking down an expression into simpler components, called factors, that when multiplied together give back the original expression. It is a fundamental technique in algebra and is often used to simplify complex expressions. In the case of our exercise, the expression was given as two separate terms with a common component, \((x+y)^2\). By recognizing this shared piece, you can factor it out.

• This simplifies the work involved, letting you combine coefficients directly.
• Essentially, factorizing means "taking out" this common part from each product.

Factorization is handy, not just for simplifying expressions, but also for solving equations. It can help in finding roots and understanding the structure of algebraic expressions. Always look for patterns like squares, cubes, or common groups in sequences to identify opportunities to factor an expression.

Distributive Property

The distributive property is key in algebra for handling expressions with parentheses. It allows you to multiply a term outside the parentheses by each term inside the parentheses. This property makes it easier to simplify expressions and solve equations. Consider an expression like \( a(b+c) \). Using the distributive property, you expand it to:

• \( ab + ac \)

In our exercise, we distributed the constants 2 and 3 to the terms within \((x+y)^2\). This created a new form but maintained the equality of the expression:

• Initially, we apply \(2 \times (x+y)^2\) and \(3 \times (x+y)^2\).

Rather than handling them as separate products, recognize the shared factor of \((x+y)^2\). The distributive property lets us work with based coefficients, easing further simplification.

Common Factors

Common factors in algebraic expressions are terms that appear in all components of an expression. Recognizing these allows you to simplify expressions or solve equations more efficiently. In our example, \((x+y)^2\) is the common factor across both terms \(2(x+y)^2\) and \(3(x+y)^2\). By treating this shared factor as a single unit, you can simplify the expression effectively.

• Identify shared elements - like \((x+y)^2\) here.
• Once identified, factor them out to reduce the complexity of the expression.

After factoring out the common term, what's left are coefficients that can be added or multiplied as needed. In this procedure, understanding and utilizing common factors greatly simplifies the calculation and provides a clearer view of the expression's structure.

Coefficient Addition

Coefficient addition is straightforward, yet understanding it is crucial when simplifying expressions. It involves the addition of numerical multipliers of variables or terms. Consider our exercise after factoring: we were left with coefficients 2 and 3 for the common factor \((x+y)^2\). The next step was simple addition: 2 + 3 = 5.

Adding coefficients reduces the expression to a more compact form, helping not only in simplification of terms but also in preparing it for further evaluations or solving.

• Always add coefficients that relate to the same variable or term. In our case, that's why 2 and 3 were added instead of modifying the factor \((x+y)^2\).

Make sure each term in the expression aligns properly. This keeps the equation balanced and correctly simplified. In algebra, being meticulous with coefficients and their addition is key to achieving an accurate final expression.