Problem 67 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \left(d^{4}+\frac{1}{4}\right)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(d^{8} + \frac{1}{2}d^{4} + \frac{1}{16}\).

Step 1: Understand the Expression

The problem requires you to perform the operations on the expression \( \left(d^{4} + \frac{1}{4}\right)^{2} \). This is a binomial raised to the second power.

Step 2: Apply the Binomial Theorem

The binomial square can be expanded using the formula: \((a+b)^{2} = a^{2} + 2ab + b^{2}\). Here, \(a = d^{4}\) and \(b = \frac{1}{4}\).

Step 3: Square the First Term

Calculate \(a^{2} = (d^{4})^{2} = d^{8}\).

Step 4: Calculate the Product of Terms and Double It

Calculate \(2ab = 2 \cdot d^{4} \cdot \frac{1}{4} = \frac{1}{2}d^{4}\).

Step 5: Square the Second Term

Calculate \(b^{2} = \left(\frac{1}{4}\right)^{2} = \frac{1}{16}\).

Step 6: Combine All Parts

Add the results from the previous steps: \(d^{8} + \frac{1}{2}d^{4} + \frac{1}{16}\). This gives the fully expanded expression.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool. It allows you to expand expressions raised to a power, often making calculations easier to handle. Specifically, for any binomial expression of the form \((a+b)^n\), the theorem gives us a way to express it as a sum of terms using binomial coefficients. These coefficients can be found in Pascal's Triangle or calculated directly.

• For example, to expand \((a+b)^2\), we use \(a^2 + 2ab + b^2\), applying coefficients derived from the triangle.
• This process simplifies potentially complex polynomial expansion processes into systematic steps.

By identifying the elements \(a\) and \(b\) in your expression, you can quickly transform an elevated binomial into a manageable polynomial. The Binomial Theorem is especially useful in algebra where efficiency and accuracy are paramount.

Polynomial Expansion

Polynomial expansion is the process of expressing a power of a binomial as a polynomial. When you expand something like \((d^4 + \frac{1}{4})^2\), you are transforming it into a sum of terms with integer coefficients. This technique involves systematically applying rules or theorems, like the Binomial Theorem, to each part of the expression.

• The initial expression \((d^4 + \frac{1}{4})^2\) is expanded into \(d^8 + \frac{1}{2}d^4 + \frac{1}{16}\), making each term visible and calculable.
• This conversion is crucial for further mathematical operations, integration, or solving equations.

Polynomial expansion lets us simplify expressions to facilitate easier manipulation and understanding in various algebraic scenarios.

Exponents

Exponents are a fundamental concept in mathematics. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, in the term \(d^4\), the base is \(d\), and it is multiplied by itself four times.

When dealing with expressions like \((d^4 + \frac{1}{4})^2\), understanding exponents helps in breaking down and simplifying the problem through exponential rules:

• Each term in the process, such as \((d^4)^2 = d^8\), follows specific rules for multiplying exponents.
• This understanding aids in the effective application of polynomial expansion and the binomial theorem.

Exponents streamline otherwise lengthy multiplication processes, turning complex solutions into workable expressions.