Solution:** To find the remainder when \( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 \) is divided by \( x - 2 \), we use the Remainder Theorem, which states that the remainder of a polynomial \( P(x) \) divided by \( x - c \) is \( P(c) \).

["# Finding the Remainder When ( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 ) Is Divided by ( x - 2 ) Using the Remainder Theorem", "When dividing a polynomial ( P(x) ) by a linear divisor ( x - c ), determining the remainder efficiently can save time and eliminate complex long division steps. One of the most powerful tools for this task is the Remainder Theorem, a fundamental concept in algebra that simplifies calculations significantly. In this article, we’ll explore how to find the remainder when ( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 ) is divided by ( x - 2 ), using the Remainder Theorem.", "## What Is the Remainder Theorem?", "The Remainder Theorem states that if a polynomial ( P(x) ) is divided by ( x - c ), the remainder is ( P(c) ). This means instead of performing polynomial long division, we simply evaluate the polynomial at ( x = c ) to find the remainder directly.", "Mathematically:\n[\n\ ext{Remainder} = P(c)\n]", "This quick method avoids time-consuming calculations and is especially valuable in algebra, calculus, and polynomial analysis.", "## Step-by-Step: Finding the Remainder", "Let’s apply the Remainder Theorem to find the remainder when dividing ( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 ) by ( x - 2 ).", "Here, ( c = 2 ). So we compute:\n[\nP(2) = (2)^4 - 5(2)^3 + 6(2)^2 + 4(2) - 8\n]", "Now calculate each term step by step:", "- ( (2)^4 = 16 )\n- ( -5(2)^3 = -5 \ imes 8 = -40 )\n- ( 6(2)^2 = 6 \ imes 4 = 24 )\n- ( 4(2) = 8 )\n- Constant term: ( -8 )", "Now sum them:\n[\nP(2) = 16 - 40 + 24 + 8 - 8 = 0\n]", "So,\n[\n\ ext{Remainder} = P(2) = 0\n]", "## Interpretation and Conclusion", "The remainder is 0, which means ( x - 2 ) is not just a divisor but a factor of ( P(x) ). In mathematical terms, ( x = 2 ) is a root of the polynomial.", "This result aligns with synthetic division or actual polynomial factoring, confirming that:\n[\nx^4 - 5x^3 + 6x^2 + 4x - 8 = (x - 2)Q(x)\n]\nfor some polynomial ( Q(x) ).", "Using the Remainder Theorem, we quickly found:\n- The remainder is ( 0 )\n- ( x - 2 ) divides ( P(x) ) exactly\n- Critical insight for further factoring or root analysis", "## Why Use the Remainder Theorem?", "- Simplicity: Evaluating at one point instead of performing full long division.\n- Speed: Saves time, especially useful in exams or multiple problem-solving steps.\n- Verification Platform: Confirms whether ( x - c ) is a factor before deeper factoring.\n- Foundational Tool: Essential for understanding polynomial behavior, graphing, and solving equations.", "## Final Thoughts", "When tackling polynomial division, the Remainder Theorem is a must-know strategy. For ( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 ), applying this principle reveals that ( x - 2 ) divides cleanly, yielding a remainder of 0 — a key insight for factorization and root testing.", "Mastering this method strengthens algebraic fluency and enables efficient problem-solving across math disciplines.", "---", "### Key Search Keywords:\n- Remainder Theorem\n- Polynomial division remainder\n- Find remainder when dividing by ( x - 2 )\n- Quick remainder calculation\n- Algebra tip for polynomial division", "### Related Articles:\n- How to Apply the Remainder Theorem\n- Polynomial division without long division\n- Using Synthetic Division with the Remainder Theorem"]