Show that the binomial can be represented as the difference of two squares.

*x*4 − 1

To show that the binomial \(x^4 - 1\) can be represented as the difference of two squares, let's proceed step by step.

### Step 1: Recall the Difference of Squares Formula
The difference of squares formula is:

\[
a^2 - b^2 = (a - b)(a + b)
\]

### Step 2: Identify \(a^2\) and \(b^2\)
We want to express \(x^4 - 1\) in the form \(a^2 - b^2\). Notice that \(x^4\) can be written as \((x^2)^2\), and \(1\) can be written as \(1^2\). So, we can write:

\[
x^4 - 1 = (x^2)^2 - 1^2
\]

### Step 3: Apply the Difference of Squares Formula
Now, apply the difference of squares formula where \(a = x^2\) and \(b = 1\):

\[
x^4 - 1 = (x^2 - 1)(x^2 + 1)
\]

### Step 4: Simplify Further (Optional)
Notice that \(x^2 - 1\) is itself a difference of squares, since:

\[
x^2 - 1 = (x - 1)(x + 1)
\]

So, the original expression can be further factored as:

\[
x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)
\]

Thus, \(x^4 - 1\) can be represented as the difference of two squares, initially as:

\[
x^4 - 1 = (x^2 - 1)(x^2 + 1)
\]

and further factored into:

\[
x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)
\]

Would you like to know more details or have any questions? Here are some related problems you might find interesting:

1. Factor \(x^4 - 16\) using the difference of squares.
2. Express \(x^6 - 1\) as a difference of squares.
3. Factor \(x^2 - 4x + 4 - y^2\) using the difference of squares.
4. Simplify \((x^2 + 1)(x - 1)(x + 1)\).
5. Show that \(x^8 - 1\) can be expressed as the difference of squares.

**Tip:** Always look for opportunities to apply the difference of squares formula when factoring expressions that involve even powers or constants like 1 or 4.

Learn how to express x^4 - 1 as the difference of two squares with a detailed step-by-step solution. This problem covers algebraic concepts such as polynomials, factoring, and the difference of squares formula. Suitable for advanced high school students.

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