Thus, the remainder when $ p(x) $ is divided by $ x^4 - 1 $ is simply $ p(x) $, since $ \deg p < 4 $. So the remainder is:

["Understanding the Remainder When Dividing Polynomials: Why $ \deg p < 4 $ Means $ p(x) $ Is Its Own Remainder Modulo $ x^4 - 1 $", "When dividing one polynomial by another, particularly in systems like polynomial modulo arithmetic, understanding the remainder is essential. A common scenario in polynomial division is when the degree of the dividend is less than the degree of the divisor. Today, we explore a key concept: if $ \deg(p(x)) < 4 $, then $ x^4 - 1 $ divides evenly into $ p(x) $, leaving $ p(x) $ as the remainder — with a crucial interpretation.", "---", "### The Divisor: $ x^4 - 1 $", "The polynomial $ x^4 - 1 $ factors nicely as:", "$$\nx^4 - 1 = (x - 1)(x + 1)(x^2 + 1)\n$$", "This divisor represents a classical boundary in polynomial arithmetic. It’s degree 4, so any polynomial $ p(x) $ with degree less than 4 cannot be further divided — meaning $ p(x) $ leaves no remainder when divided by $ x^4 - 1 $.", "---", "### The Remainder in Polynomial Division", "By the Polynomial Division Algorithm, any polynomial $ p(x) $ divided by a divisor $ d(x) $ yields:", "$$\np(x) = q(x) \cdot d(x) + r(x)\n$$", "where $ \deg(r) < \deg(d) $. Here, $ \deg(d(x)) = 4 $, so the remainder $ r(x) $ must satisfy $ \deg(r) < 4 $. Typical algorithms (like long division) produce a unique remainder of degree less than 4 — often a cubic or lower polynomial.", "---", "### But Why Is $ p(x) $ Itself the Remainder When $ \deg p < 4 $?", "The catch lies in interpreting the division modulo $ x^4 - 1 $. In modular arithmetic, when $ \deg(p) < 4 $, $ p(x) $ is congruent to itself modulo $ x^4 - 1 $, because no division occurs — the remainder cannot be smaller than zero degree. Formally:", "$$\np(x) \equiv p(x) \mod (x^4 - 1)\n$$", "That is, $ p(x) $ is already the remainder by definition when the degree of $ p $ is less than the degree of the divisor. No further subtraction is needed.", "---", "### Applications and Insights", "This insight simplifies computations in various mathematical and computational contexts:", "- In finite field constructions and error-correcting codes, reducing polynomials modulo $ x^4 - 1 $ is routine, especially when working with signals or polynomials of bounded degree.\n- In algorithm design, recognizing that inputs of degree < 4 directly yield remainders avoids costly division steps.\n- From an algebraic perspective, this reinforces how degree bounds govern remainder behavior — a fundamental principle in abstract algebra and number theory.", "---", "### Conclusion", "Thus, when $ \deg(p(x)) < 4 $, dividing $ p(x) $ by $ x^4 - 1 $ yields:", "$$\n\ ext{Remainder} = p(x)\n$$", "This behavior reflects the core property of polynomial division modulo $ x^4 - 1 $: any polynomial of lesser degree is its own equivalence class, and no genuine reduction is required. Understanding this clarifies many polynomial computations and strengthens foundational knowledge in algebra and applied mathematics.", "---", "Key Takeaway:\nFor any polynomial $ p(x) $ with $ \deg p < 4 $, the remainder when divided by $ x^4 - 1 $ is $ p(x) $ — not due to complexity, but because degree constraints define the remainder uniquely.", "---", "Keywords: remainder $ p(x) $, division $ x^4 - 1 $, polynomial remainder, degree less than 4, $ p(x) \mod (x^4 - 1) $, polynomial arithmetic, algebra foundation."]