Problem 58 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \left(8 x^{4}-3\right)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \( 64x^8 - 48x^4 + 9 \).

Step 1: Understand the Problem

The given expression \[ (8x^4 - 3)^2 \]must be expanded using the formula for the square of a binomial, \[ (a-b)^2 = a^2 - 2ab + b^2. \] Identify \( a = 8x^4 \) and \( b = 3 \).

Step 2: Square the First Term

Square the first term \( a = 8x^4 \) to get\[ (8x^4)^2 = 64x^8. \]

Step 3: Compute the Double Product of the Terms

Calculate the product of \( 2 \), \( a = 8x^4 \), and \( b = 3 \):\[ 2 imes 8x^4 imes 3 = 48x^4. \] Since this is subtracted in the formula, it becomes:\[ -48x^4. \]

Step 4: Square the Second Term

Now, square the second term \( b = 3 \) to get\[ 3^2 = 9. \]

Step 5: Assemble the Expanded Form

Bring all results together to expand the expression:\[ (8x^4 - 3)^2 = 64x^8 - 48x^4 + 9. \]

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a cornerstone in mathematics, representing combinations of numbers, variables, and operations. These expressions are formed using arithmetic operations such as addition, subtraction, multiplication, and division. In the given exercise, the expression \((8x^4 - 3)^2\) involves a combination of variables and constants. Here:

• \(8x^4\) is a term where 8 is a coefficient, \(x\) is a variable, and 4 is the exponent showing the power to which \(x\) is raised.
• The number 3 is a constant term.

This expression is a binomial because it contains exactly two terms. Understanding algebraic expressions requires recognizing each element's role and how they can be manipulated through operations to expand or simplify.

Squaring Binomials

Squaring a binomial is a mathematical operation where we raise a two-term algebraic expression to the second power. This involves the binomial formula \((a-b)^2 = a^2 - 2ab + b^2\). The concept comes into play when expanding expressions like \((8x^4 - 3)^2\). To apply this:

• Identify the terms: \(a = 8x^4\) and \(b = 3\).
• First, square \(a\), which is \((8x^4)^2 = 64x^8\).
• Next, calculate the product of the double of \(a\) and \(b\), \(-2ab = -48x^4\).
• Finally, square \(b\), which results in \(9\).

When squaring binomials, follow the formula to arrive at the expanded polynomial and combine all parts: \(64x^8 - 48x^4 + 9\). This process demonstrates how binomials transform into polynomials.

Performing Polynomial Operations

Polynomial operations encompass a variety of actions performed on polynomials, such as addition, subtraction, multiplication, and even raising expressions to powers. In the case of the exercise \((8x^4 - 3)^2\), polynomial operations involve expanding the expression into its simplest form. Here's how:

• Compute each term individually by following the binomial expansion formula.
• Ensure proper handling of coefficients and exponents, such as calculating \(64x^8\) from \((8x^4)^2\).
• Subtract and add terms like the double product \(-48x^4\) and the square of the constant 9.

Understanding polynomial operations is crucial as it helps simplify and organize expressions, making them easier to solve or further manipulate. Mastering these operations builds a strong foundation for algebra and higher-level mathematics.