Problem 66 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (4 a+2)^{2} $$

Step-by-Step Solution

Answer

The expanded form of \((4a+2)^2\) is \(16a^2 + 16a + 4\).

Step 1: Identify and Expand the Squaring

We start by recognizing that we need to expand \((4a + 2)^2\), which is a perfect square trinomial.

Step 2: Apply the Expansion Formula

To expand \((4a + 2)^2\), use the formula for squaring a binomial: \((x + y)^2 = x^2 + 2xy + y^2\). Here, \(x = 4a\) and \(y = 2\).

Step 3: Calculate the Individual Terms

Substitute in our binomial: \((4a)^2 = 16a^2\), \(2 \times 4a \times 2 = 16a\), and \(2^2 = 4\).

Step 4: Combine the Results

Put the calculated values together to form the final expanded expression: \(16a^2 + 16a + 4\).

Key Concepts

Binomial Expansion

The binomial expansion is a technique used in algebra to expand expressions that are raised to a power. Understanding it allows us to simplify expressions like \((x + y)^n\). When we expand \((4a + 2)^2\), we utilize the concept of binomial expansion. It's important to recognize that this involves two terms, in this case, \(4a\) and \(2\), raised to a power, which is 2.
Simply put, binomial expansion helps break down expressions into simpler parts.

• Identify the binomial: Recognize parts of the expression that need expansion. Here, our binomial is \((4a + 2)\).
• Apply the binomial square formula: The formula \((x + y)^2 = x^2 + 2xy + y^2\) is used to find the expanded form.
• Calculate individual terms: Substitute and compute each part separately, combining them at the end.

This method simplifies solving and working with algebraic expressions.

Perfect Square Trinomial

A perfect square trinomial arises when you square a binomial and end up with a three-term polynomial. It's a significant concept in algebra because it allows us to predict and form neat polynomial expressions like \(x^2 + 2xy + y^2\). When expanding \((4a + 2)^2\), we use the structure of a perfect square trinomial:

• First term squared: Calculate \((4a)^2\) to get \(16a^2\).
• Double the product of the terms: Find \(2 \times 4a \times 2 = 16a\).
• Second term squared: Compute \(2^2\) to get \(4\).

The culmination of these calculations gives us \(16a^2 + 16a + 4\), neatly forming a perfect square trinomial.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. They are foundational to algebra and more advanced mathematics. Understanding and manipulating these expressions, like \((4a + 2)^2\), involves recognizing the variables and constants involved along with the appropriate operations.

• Variables: Symbols like \(a\) that represent numbers whose values may change.
• Constants: Specific numbers like 2 in our expression that do not change.
• Operations: Actions such as addition, subtraction, multiplication, and division that are applied to variables and constants.

By becoming fluent in the basics of algebraic expressions, students can solve more complex problems and recognize patterns, leading to mastery in algebra.