What is the remainder when \( x^4 - 3x^2 + 2x + 5 \) is divided by \( x - 1 \)?
["Title: What is the Remainder When ( x^4 - 3x^2 + 2x + 5 ) is Divided by ( x - 1 )?", "Meta Description:\nDiscover the remainder when ( x^4 - 3x^2 + 2x + 5 ) is divided by ( x - 1 ). Learn how the Remainder Theorem simplifies polynomial division and find the exact value using substitution.", "---", "### Understanding Polynomial Division and the Remainder Theorem", "When dividing a polynomial by a linear divisor of the form ( x - a ), special insight from the Remainder Theorem helps avoid lengthy long division. According to this theorem, the remainder of dividing a polynomial ( f(x) ) by ( x - a ) is simply ( f(a) ).", "This simple formula saves time and effort, especially with higher-degree polynomials like ( x^4 - 3x^2 + 2x + 5 ). Instead of performing full polynomial long division, we evaluate the polynomial at ( x = 1 )—the value that makes the divisor ( x - 1 = 0 ).", "---", "### Applying the Remainder Theorem to the Given Polynomial", "Let ( f(x) = x^4 - 3x^2 + 2x + 5 ).\nTo find the remainder when ( f(x) ) is divided by ( x - 1 ), compute:", "[\nf(1) = (1)^4 - 3(1)^2 + 2(1) + 5\n]", "Calculate step by step:\n- ( 1^4 = 1 )\n- ( -3(1)^2 = -3 )\n- ( 2(1) = 2 )\n- Constant term = 5", "Now, sum the values:\n[\nf(1) = 1 - 3 + 2 + 5 = 5\n]", "---", "### Why This Works: A Quick Explanation", "The Remainder Theorem leverages the fact that dividing by ( x - a ) removes one degree of ( x ), leaving a constant remainder. When ( x = a ), the polynomial evaluates exactly to that constant. This method applies to all polynomials and avoids complex division steps.", "---", "### Final Answer", "The remainder when ( x^4 - 3x^2 + 2x + 5 ) is divided by ( x - 1 ) is 5.", "---", "### Applications and Benefits", "Knowing how to compute remainders efficiently using substitution:\n- Speeds up problem-solving in algebra\n- Supports more advanced topics like factor theorems and roots\n- Helps in designing numerical algorithms for polynomial evaluation", "---", "Keywords: remainder when dividing x⁴ - 3x² + 2x + 5 by x - 1, Remainder Theorem, polynomial division, substitute x = 1, algebraic methods, high school algebra, math tutoring tips.", "---", "Note: Use this approach anytime you need the remainder of a polynomial division—just plug in ( x = a ) into the polynomial!"]
