Problem 34 - Elementary Algebra

Question

For the following problems, find the products. $$ \left(a+\frac{1}{2}\right)^{2} $$

Step-by-Step Solution

Answer

Answer: The product of the given expression is \(a^2 + a + \frac{1}{4}\).

Step 1: Identify the binomial expression

The given expression is a binomial of the form \(a+b\), where \(a=a\) and \(b=\frac{1}{2}\).

Step 2: Apply the binomial expansion formula

We will use the formula \((a+b)^2 = a^2 + 2ab + b^2\) with our values of \(a\) and \(b\). Substituting our values of \(a\) and \(b\) into the formula, we get: $$ \left(a+\frac{1}{2}\right)^{2} = a^2 + 2\times a\times \frac{1}{2} + \left(\frac{1}{2}\right)^2 $$

Step 3: Simplify the expression

Now, let's simplify the expression step by step: $$ = a^2 + a + \frac{1}{4} $$ So, the product of the given expression is: $$ \left(a+\frac{1}{2}\right)^{2} = a^2 + a + \frac{1}{4} $$

Key Concepts

Algebra

Algebra is a significant branch of mathematics that deals with symbols and the rules for manipulating these symbols. It provides a way to write formulas and solve equations using mathematical expressions. In our example, the expression \(\left(a+\frac{1}{2}\right)^{2}\) involves the use of variables and constants. Here, the variable is \(a\) and the constant is \(\frac{1}{2}\).
When working with algebra:

• Understand that variables represent numbers and can change.
• Apply arithmetic operations upon the variables and constants.
• Utilize algebraic expressions to model real-life situations.

Algebra is essential as it forms the basis for more advanced topics, helping simplify complex problems, and offering solutions in a structured manner.

Binomial Theorem

The binomial theorem is a powerful tool in algebra, offering a way to expand expressions that are raised to a power. A binomial is a simple algebraic expression comprising two terms, like \(a + b\). The binomial theorem gives us a formula for expanding expressions like \((a + b)^n\). For smaller powers like \((a + b)^2\), it is a shortcut to avoid manual multiplication.
In our solution:

• The formula used is \((a+b)^2 = a^2 + 2ab + b^2\) which simplifies the expansion process.
• By substituting \(a\) and \(b\) from the expression \((a+\frac{1}{2})^2\), calculations become straightforward and efficient.

Understanding the binomial theorem allows us to manage complex algebraic expansions with ease, making it an essential concept for both basic and more advanced levels of study.

Polynomials

Polynomials are expressions involving a sum of powers of one or more variables multiplied by coefficients. They can be simple, like \(a\), or more complex, like \(a^2 + a + \frac{1}{4}\), where multiple terms are combined.
Some key aspects include:

• Each term in a polynomial consists of a coefficient and a variable raised to a power.
• Polynomials are classified by degree, which is the highest power of the variable in the expression.
• In our given expression, \(a^2 + a + \frac{1}{4}\), the degree is 2, indicating a quadratic polynomial.

Polynomials are important in various fields such as physics, engineering, and economics, as they often represent relationships and trends. Recognizing and manipulating polynomial expressions is a fundamental skill in algebra, aiding in problem-solving and analytical thinking.