Problem 106 - Algebra A Combined Function

Question

Perform each indicated operation. \((2 x+1)^{2}\)

Step-by-Step Solution

Answer

\((2x+1)^2 = 4x^2 + 4x + 1\).

Step 1: Identify the Operation

The exercise requires expanding the expression \((2x+1)^2\). This involves using the identity \((a+b)^2 = a^2 + 2ab + b^2\) where \(a = 2x\) and \(b = 1\).

Step 2: Expand the Square

Using the formula from Step 1, expand \((2x+1)^2\) as follows: - The first term \(a^2\) is \((2x)^2 = 4x^2\).- The second term \(2ab\) is \(2\cdot(2x)\cdot1 = 4x\).- The third term \(b^2\) is \(1^2 = 1\).

Step 3: Combine All Terms

Combine the terms derived in Step 2: \[ (2x+1)^2 = 4x^2 + 4x + 1 \]

Key Concepts

Expanding Expressions

Expanding expressions involves breaking down mathematical expressions into simpler forms. It often requires utilizing algebraic identities to make the process faster and easier. When you encounter expressions like \((2x + 1)^2\), which involve a power, expansion can be a handy tool. Break the expression \((a + b)^2\) into the identity \(a^2 + 2ab + b^2\). This means you will take each component of the binomial, apply the identity, and calculate each term step by step.

During expansion, you:

• Square each component of the binomial separately.
• Calculate the cross-product, which is \(2ab\).
• Add all the resulting terms together.

This straightforward process breaks the expression down, helping in simplifying and solving equations or problems involving polynomials.

Binomial Theorem

The binomial theorem is a significant mathematical technique used to expand expressions raised to a power. It applies particularly to expressions like \((a + b)^n\). The theorem provides a formula that is useful when tackling problems, as it simplifies the task of expansion.

For the case of \((a + b)^2\), the theorem gives us:

• First, compute \(a^2\), which is the square of the first term.
• Next, calculate \(2ab\), which represents two times the product of both terms.
• Finally, calculate \(b^2\), which is the square of the second term.

This process helps capture all interactions between the terms in the binomial, and as a result, provides a simple set of terms that represent the expanded expression. With more complex powers and combinations, the binomial theorem extends further, adding terms based on the specific power \(n\). Using the binomial theorem facilitates accurate and efficient solutions of binomial expansions.

Mathematical Identities

Mathematical identities are tools in algebra that simplify complex expressions or calculations. They define a basic truth or equation that holds for all values of the variables within them. One powerful identity used in expanding squares is \((a + b)^2 = a^2 + 2ab + b^2\). This identity makes it easy to quickly expand expressions without extensive computation.

In practice, identities serve as shortcuts:

• They provide a reliable pattern or formula to follow when expanding or simplifying algebraic expressions.
• Using identities helps in reducing errors, as they are universally true and derived from basic algebraic principles.
• They are essential in problem-solving, allowing you to connect different parts of mathematics with ease.

Knowing and applying these identities can significantly enhance your capability in working through algebraic problems, ensuring efficient and accurate results.