Problem 53 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((2 u+v)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(4u^2 + 4uv + v^2\).

Step 1: Identify the Appropriate Special Product Formula

The expression \((2u + v)^2\) resembles the square of a binomial. We will use the formula \((a + b)^2 = a^2 + 2ab + b^2\) to expand and simplify it.

Step 2: Identify the Terms of the Expression

In the expression \((2u + v)^2\), identify \(a = 2u\) and \(b = v\). These terms will be used in the expansion formula.

Step 3: Apply the Special Product Formula

Substitute \(a = 2u\) and \(b = v\) into the formula \((a + b)^2 = a^2 + 2ab + b^2\). This gives us: \((2u)^2 + 2(2u)(v) + v^2\).

Step 4: Calculate Each Term

Calculate each term of the expanded expression: - \((2u)^2 = 4u^2\)- \(2(2u)(v) = 4uv\)- \(v^2 = v^2\)

Step 5: Write the Expanded Expression

Combine all terms to write the fully expanded expression: \(4u^2 + 4uv + v^2\).

Key Concepts

Special Product Formula

The special product formulas are a set of algebraic rules that help us simplify and expand expressions more efficiently. They are especially useful when dealing with binomials. Instead of multiplying each term separately, these formulas provide a shortcut.
- For squaring a binomial like \[(a + b)^2\], the formula is: \[ a^2 + 2ab + b^2 \].
- By using this formula, you can quickly expand any expression of the form \[(a + b)^2\] without having to do each multiplication step-by-step.
Understanding and applying these formulas can save time and effort, especially in more complex algebraic problems. The use and benefits of special product formulas make them an essential tool in algebra.

Square of a Binomial

The concept of squaring a binomial involves multiplying a binomial by itself. A binomial is an algebraic expression with two terms, such as \(2u + v\). When you square a binomial, the goal is to find the expanded form of the expression:

• Identify each term in the binomial. In our example, \(a = 2u\) and \(b = v\).
• Apply the square of a binomial formula: \[(a + b)^2 = a^2 + 2ab + b^2\].
• Substitute the identified terms into the formula.

This gives you an easier way to expand without manually multiplying the binomial twice. By memorizing this pattern, you can find the expanded forms quickly and accurately.

Expanding Binomials

Expanding binomials is the process of breaking down an expression in the form \((a + b)^n\) into a simplified expression where all terms are fully expanded.
In the case of \((2u + v)^2\), expansion involves using the formula to simplify the expression:

• Calculate \((2u)^2\), which results in \(4u^2\).
• Multiply \(2 \cdot 2u \cdot v\), resulting in \(4uv\).
• Add \(v^2\) to complete the expansion.

The resulting expression, \(4u^2 + 4uv + v^2\), is the fully expanded form of \((2u + v)^2\).
Understanding how to expand binomials is crucial because it simplifies solving polynomial equations and working with complex algebraic expressions.