Solution:** To find the remainder when \( f(x) \) is divided by \( x - 1 \), we use the Remainder Theorem, which states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \). Here, \( c = 1 \), so we calculate \( f(1) \):

["Finding Remainders with the Remainder Theorem: A Simple Guide", "When working with polynomials, one common problem is determining the remainder when a polynomial ( f(x) ) is divided by a linear divisor like ( x - c ). Fortunately, the Remainder Theorem provides a quick and efficient solution.", "### What Does the Remainder Theorem Say?", "The Remainder Theorem states:\nIf a polynomial ( f(x) ) is divided by ( x - c ), the remainder is ( f(c) ).\nThis means you don’t need to perform long division—just substitute ( x = c ) into the polynomial function.", "### Why Is This Useful?", "Instead of carrying out costly long division, especially with high-degree polynomials, evaluating ( f(c) ) gives the exact remainder instantly. This is particularly helpful in algebra, calculus, and applied math when analyzing polynomial behavior at specific points.", "### Example: Applying the Theorem", "Consider the polynomial:\n[\nf(x) = x^3 - 4x^2 + 5x - 2\n]\nWe want to find the remainder when ( f(x) ) is divided by ( x - 1 ). According to the Remainder Theorem, compute ( f(1) ):", "[\nf(1) = (1)^3 - 4(1)^2 + 5(1) - 2\n]\n[\nf(1) = 1 - 4 + 5 - 2 = 0\n]", "Thus, the remainder is ( 0 ). This also tells us that ( x - 1 ) is a factor of ( f(x) )—which we might suspect from the result.", "### How to Use This in Real Problems", "- Root testing: A remainder of zero indicates ( c ) is a root of ( f(x) ), helping factor polynomials.\n- Function evaluation: Quickly evaluate polynomial behavior at any point without expansion.\n- Simplify complex divisions: Avoid lengthy division steps in exams or problem-solving settings.", "### Summary", "Using the Remainder Theorem to find the remainder when dividing ( f(x) ) by ( x - 1 ) is fast, accurate, and mathematically grounded. Simply substitute ( x = 1 ) into ( f(x) ), and that’s your answer—making polynomial division much simpler and more intuitive.", "Start applying the Remainder Theorem today to simplify polynomial division, confirm roots, and enhance algebraic fluency.", "Keywords: Remainder Theorem, polynomial remainder, divide by x - 1, substitute c to find remainder, math strategy, polynomial evaluation, algebra tips, open education resource."]