Let’s compute $ x^4 + 1 $, $ x^4 - 1 $:

["# Understanding Polynomial Expressions: Computing $x^4 + 1$ and $x^4 - 1$", "In algebra, working with polynomial expressions is fundamental for solving equations, analyzing functions, and building mathematical models. Two key expressions often studied are $x^4 + 1$ and $x^4 - 1$. While they appear similar, their algebraic behavior, factorizations, and applications differ significantly. In this article, we explore how to compute and interpret these polynomials, their factorizations, roots, and practical uses.", "## What Are $x^4 + 1$ and $x^4 - 1$?", "At their core, $x^4 + 1$ and $x^4 - 1$ are quartic (degree 4) polynomials. These expressions arise in various mathematical contexts, including number theory, calculus, and computer algebra. Understanding them helps deepen knowledge of polynomial factorization, roots, and symmetry.", "### $x^4 + 1$: Sum of Fourth Powers", "The expression $x^4 + 1$ is the sum of a fourth power and 1. Unlike $x^2 + a^2$, which is irreducible over the reals, $x^4 + 1$ can be factored over the complex and real numbers, though not into linear factors in real numbers alone. It plays a crucial role in cyclotomic polynomials and roots of unity.", "### $x^4 - 1$: Difference of Fourth Powers", "On the other hand, $x^4 - 1$ is a difference of squares, since it can be rewritten as $(x^2)^2 - 1^2 = (x^2 + 1)(x^2 - 1)$. This expression further factorizes into linear terms with complex coefficients: $x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)$. It is fully factored over the reals and integers.", "---", "## Factoring Techniques for $x^4 + 1$ and $x^4 - 1$", "### Factoring $x^4 - 1$", "The expression $x^4 - 1$ is a classic example for applying difference of squares:", "$$\nx^4 - 1 = (x^2 + 1)(x^2 - 1)\n$$", "Further, $x^2 - 1$ is also a difference of squares:", "$$\nx^2 - 1 = (x + 1)(x - 1)\n$$", "So, complete factorization over the reals:", "$$\nx^4 - 1 = (x - 1)(x + 1)(x^2 + 1)\n$$", "Since $x^2 + 1$ has no real roots — its discriminant is negative — this is its full real factorization.", "### Factoring $x^4 + 1$", "Unlike $x^4 - 1$, $x^4 + 1$ cannot be factored into real quadratic or linear terms. However, it factors over the complex numbers using roots of unity. The equation $x^4 + 1 = 0$ implies $x^4 = -1$, so the roots are the fourth roots of $-1$, which are:", "$$\nx = e^{i\frac{\pi}{4}},\ e^{i\frac{3\pi}{4}},\ e^{i\frac{5\pi}{4}},\ e^{i\frac{7\pi}{4}}\n$$", "These correspond to:", "$$\nx = \frac{\sqrt{2}}{2}(\pm 1 \pm i)\n$$", "Thus, $x^4 + 1$ factors as:", "$$\nx^4 + 1 = (x - e^{i\pi/4})(x - e^{i3\pi/4})(x - e^{i5\pi/4})(x - e^{i7\pi/4})\n$$", "Over the reals, this is equivalent to the product of two irreducible quadratics:", "$$\nx^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1)\n$$", "This factorization preserves real coefficients and is commonly used in polynomial algebra and signal processing.", "---", "## Roots and Graphs: Behavior Across Real and Complex Numbers", "### Real Roots", "- $x^4 - 1 = 0$ has four real roots: $x = \pm 1, \pm i$. Only $x = \pm 1$ are real.\n- $x^4 + 1 = 0$ has no real roots — its roots lie entirely in the complex plane.", "### Graphs", "Plotting $y = x^4 - 1$, we see a quartic curve passing through $(-1,0)$, $(1,0)$, and peaking at $x = 0$ with $y = -1$. The graph is symmetric about the y-axis.", "For $y = x^4 + 1$, the curve always lies strictly above $y = 1$, having a minimum at $y = 1$ when $x = 0$, and rising to infinity as $x \ o \pm\infty$. It is symmetric and minimized at $x=0$.", "---", "## Applications and Importance", "### $x^4 - 1$", "Used in:", "- Polynomial division and remainder theorem\n- Cyclotomic polynomials and number theory\n- Finite differences and signal analysis\n- Factoring exercises in algebra\n- Modeling real-world systems with symmetric roots (e.g., mechanical systems)", "### $x^4 + 1$", "Significant in:", "- Complex analysis and complex roots\n- Fourier transforms and spectral theory\n- Cryptography and error-correcting codes\n- Quadratic forms and number theory\n- Signal processing as an irreducible quartic in transform domains", "---", "## Conclusion", "While both $x^4 + 1$ and $x^4 - 1$ represent quartic polynomials, their algebraic behaviors differ fundamentally. $x^4 - 1$ is easily factorable over the reals and opens a window into real roots and symmetry, whereas $x^4 + 1$ reveals deeper connections to complex analysis and roots of unity. Understanding these expressions enhances algebraic intuition and prepares learners for advanced topics in mathematics, engineering, and computer science.", "---", "### Summary Table", "| Aspect | $x^4 - 1$ | $x^4 + 1$ |\n|--------|------------|------------|\n| Type | Difference of squares | Sum of squares (irreducible over reals) |\n| Real Roots | $x = \pm 1$ | None |\n| Factorization (Reals) | $(x - 1)(x + 1)(x^2 + 1)$ | $(x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1)$ |\n| Complex Roots | $\pm1, \pm i$ | $e^{i\pi/4}, e^{i3\pi/4}, e^{i5\pi/4}, e^{i7\pi/4}$ |\n| Graph Behavior | Symmetric, crosses x-axis | Symmetric, minimum at $y=1$ |", "---", "Keywords: $x^4 + 1$ computation, $x^4 - 1$ factorization, polynomial expressions, algebraic factorization, complex roots, real roots, cyclotomic polynomials, polynomial algebra, mathematical foundations."]