Problem 87 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(2 x+1)^{2}$$

Step-by-Step Solution

Answer

The expanded expression is \(4x^2 + 4x + 1\).

Step 1: Identify the Expression

The expression given is \((2x+1)^2\). This means we're taking \((2x+1)\) and multiplying it by itself.

Step 2: Apply the Binomial Square Formula

The square of a binomial \((a+b)^2\) is expanded as \(a^2 + 2ab + b^2\). In our expression, \(a = 2x\) and \(b = 1\).

Step 3: Calculate \(a^2\)

Find \(a^2\) by squaring \(2x\). Thus, \((2x)^2 = 4x^2\).

Step 4: Calculate \(2ab\)

Find \(2ab\) by multiplying \(2 imes (2x) imes 1 = 4x\).

Step 5: Calculate \(b^2\)

Find \(b^2\) by squaring \(1\). Thus, \(1^2 = 1\).

Step 6: Combine the Results

Add the results from Steps 3, 4, and 5 together: \(4x^2 + 4x + 1\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions that are raised to a power. When you have a binomial, which is a sum of two terms like - \((a + b)\) and you want to raise it to a power, the Binomial Theorem provides the formula for expansion. For square binomials, the formula simplifies to:- \((a+b)^2 = a^2 + 2ab + b^2\).This tells us that the square of a sum is composed of three parts:

• The square of the first term \(a^2\),
• Twice the product of the two terms \(2ab\),
• The square of the second term \(b^2\).

In our exercise with \((2x+1)^2\), we applied this formula to get the expanded expression. Identifying each part is crucial. Here, we set \(a = 2x\) and \(b = 1\). By expanding correctly, we simplify the process of multiplication, making complex problems more manageable.

Polynomial Multiplication

Polynomial multiplication involves varying techniques depending on the expressions involved. It's essentially multiplying every term of one polynomial by every term of another. In our case with \((2x + 1)^2\), we're using a specific type known as the square of a binomial. The procedure is:

1. Identify terms to be multiplied: Here we consider \((2x + 1) \times (2x + 1)\).
2. Apply distributive property: Each term in the first binomial multiplies each term in the second one, resulting in \(4x^2\), \(4x\), and \(1\).

Each multiplication step follows the rules of exponents and coefficients, which means:

• Multiply coefficients (numbers) together.
• Add exponents for the same variables.

It's systematic and straightforward, especially with practice. As seen, polynomial multiplication underpins operations like factoring and expanding, critical for solving quadratic expressions.

Algebraic Expansion

Algebraic expansion turns expressions from a compact, multiplicative form into an additive one, revealing the clear parts of equations or expressions. It involves breaking down a binomial, like \((2x+1)^2\), into its individual components. By expanding, each term of the binomial is squared, multiplied, and summed respectively. This makes it easier to handle within equations by clearly showing each component.Here’s how it happens for our example:- Identify initial expression: \((2x+1)^2\).- Use the binomial square formula:\(4x^2 + 4x + 1\).The resulting terms are now separate and manageable. With expansion, complex expressions can be rearranged and organized, facilitating simplification or solving within larger algebraic contexts. This operation forms the cornerstone of algebraic manipulation and understanding; it’s about moving from a closed form to an open and approachable structure.