Problem 31 - Precalculus Mathematics for Calculus

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 Special Product Formula

The expression \((2u + v)^2\) is a perfect square trinomial. It can be expanded using the square of a binomial formula: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2u\) and \(b = v\).

Step 2: Calculate the Square of the First Term

Use the formula and calculate \((2u)^2 = 4u^2\). This is the first term in the expanded expression.

Step 3: Calculate the Double Product of Both Terms

Calculate the middle term using the formula: \(2 \times 2u \times v = 4uv\).

Step 4: Calculate the Square of the Second Term

Use the formula to find the square of the second term: \(v^2\).

Step 5: Combine All Terms

Substitute the calculated terms into the formula: \((2u + v)^2 = 4u^2 + 4uv + v^2\). This gives the final simplified expression.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of algebraic expression that arises from squaring a binomial. This expression has exactly three terms. When you see the square of a binomial, it can always be expanded into a perfect square trinomial. For example, the binomial \((a + b)^2\) expands to the trinomial \(a^2 + 2ab + b^2\).

• The first term \(a^2\) is the square of the first part of the binomial.
• The second term \(2ab\) accounts for twice the product of both binomial parts.
• The third term \(b^2\) is simply the square of the second part of the binomial.

When applied to \((2u + v)^2\), it results in the trinomial \(4u^2 + 4uv + v^2\). Each of these components plays a specific role in forming the perfect square trinomial.

Binomial Expansion

Binomial expansion involves converting an expression like \((a + b)^n\) into a sum of multiple terms without parentheses. The process uses special product formulas to simplify expressions given that \(n\) is typically a small whole number. In the context of \((a+b)^2\), binomial expansion helps us systematically break down the expression into:

• \(a^2\) for the first term squared,
• \(2ab\) as the middle term from multiplying the terms together twice,
• and \(b^2\) for the square of the last term.

Utilizing binomial expansion ensures each term is accounted for and provides an organized way to solve and simplify binomials like \((2u+v)^2\) into individual terms \(4u^2 + 4uv + v^2\). This simplifies understanding and computation, particularly in algebraic contexts.

Square of a Binomial

The square of a binomial is one of the most recognizable special products in algebra. It refers to multiplying a two-term expression, or binomial, by itself. The formula \((a + b)^2 = a^2 + 2ab + b^2\) is applied when squaring a binomial. This expansion involves three key components:

• First term squared: Square the initial term \(a\) to get \(a^2\).
• Double product of the terms: Calculate \(2ab\) by multiplying the two terms together and doubling the result.
• Second term squared: Finally, square the second term \(b\) to get \(b^2\).

For \((2u + v)^2\), substituting \(a = 2u\) and \(b = v\) into this formula gives the expression \(4u^2 + 4uv + v^2\). The understanding of a square of a binomial simplifies working with quadratic expressions and is a powerful tool for many algebraic operations.