Problem 75 - Algebra for College Students

Question

Find the indicated products. Assume all variables that appear as exponents represent positive integers. $$\left(x^{n}-4\right)\left(x^{n}+4\right)$$

Step-by-Step Solution

Answer

The product is \(x^{2n} - 16\).

Step 1: Recognize the Pattern

The expression \((x^n - 4)(x^n + 4)\) fits the pattern of the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\).

Step 2: Identify 'a' and 'b' in the Pattern

In the expression \((x^n - 4)(x^n + 4)\), identify \(a = x^n\) and \(b = 4\).

Step 3: Apply the Difference of Squares Formula

Substitute \(a = x^n\) and \(b = 4\) into the formula: \((a - b)(a + b) = a^2 - b^2\). This gives \((x^n)^2 - 4^2\).

Step 4: Simplify the Exponent

Calculate \((x^n)^2\) as \(x^{2n}\).

Step 5: Compute the Square of the Constant

Calculate \(4^2\) as 16.

Step 6: Write the Final Simplified Expression

Combine the results from the previous steps: \(x^{2n} - 16\).

Key Concepts

Understanding Polynomial Multiplication

Polynomial multiplication might seem intimidating at first, but it’s all about distributing terms and combining like terms. When multiplying two polynomials, you multiply each term in the first polynomial by each term in the second. In our exercise, we multiply

• \((x^n - 4)(x^n + 4)\).

This polynomial multiplication is simplified using a special trick called the difference of squares.
The difference of squares formula helps simplify products of a special kind of binomial.
When facing two terms like

• \((a - b)(a + b)\)

the product is always

• \(a^2 - b^2\).

This trick reduces the complexity of tedious multiplication by avoiding expansion and foiling.

Exponentiation and Its Role Here

Exponentiation involves raising numbers to a power and plays a key role when dealing with algebraic expressions like polynomials. In our example, the expression incorporates exponentiation with terms such as

• \(x^n\).

Raising a power to another power, like

• \((x^n)^2\)

is solved by multiplying the exponents, which gives us

• \(x^{2n}\).

Exponents simplify writing and calculations, compactly representing large products.
This is particularly useful in simplifying terms and recognizing patterns like

• the difference of squares.

By understanding how exponentiation works, you can easily manipulate and simplify expressions to arrive at their most reduced forms.

The Role of Algebraic Expressions

Algebraic expressions form the foundation of algebra and consist of variables, coefficients, and constants, connected by arithmetic operations. In our problem, we are working with an expression that has a variable component

• \(x^n\)

and constant terms like

• 4.

This expression represents a combination of these elements.
Understanding algebraic expressions is crucial, as they allow you to generalize mathematical relationships and solve real-world problems.
The simplification process involves using known identities such as the difference of squares to refine or shorten an expression. These expressions also help in understanding more complex phenomena in math and other fields.
Moreover, recognizing how expressions are structured allows students to apply strategic simplification techniques to arrive at concise solutions.