Solution:** To verify if \( x = 1 \) is a root of multiplicity greater than 1 for \( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \), we first check if \( p(1) = 0 \).
["Solution: Verifying if ( x = 1 ) is a Root of Multiplicity Greater Than 1 for ( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 )", "Determining whether ( x = 1 ) is a multiple root of the polynomial ( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 ) is essential in understanding the nature of its zeros, especially in applications involving polynomial factorization and root behavior. A root ( x = r ) has multiplicity greater than one if ( (x - r)^2 ) divides the polynomial, or more generally, if repeated factorization reveals higher power factors at that root.", "One effective method to verify if ( x = 1 ) is a root of multiplicity greater than 1 is to check two key conditions:", "1. Does ( p(1) = 0 )?\n2. Is the derivative ( p'(1) = 0 )?", "If both conditions hold, ( x = 1 ) is at least a double root, indicating its multiplicity exceeds 1.", "---", "Step 1: Compute ( p(1) )\nSubstitute ( x = 1 ) into ( p(x) ):", "[\np(1) = (1)^4 - 4(1)^3 + 6(1)^2 - 4(1) + 1 = 1 - 4 + 6 - 4 + 1 = 0\n]", "Since ( p(1) = 0 ), ( x = 1 ) is indeed a root.", "---", "Step 2: Compute the Derivative ( p'(x) )\nNext, calculate the first derivative of ( p(x) ):", "[\np'(x) = \frac{d}{dx}(x^4 - 4x^3 + 6x^2 - 4x + 1) = 4x^3 - 12x^2 + 12x - 4\n]", "Now evaluate ( p'(1) ):", "[\np'(1) = 4(1)^3 - 12(1)^2 + 12(1) - 4 = 4 - 12 + 12 - 4 = 0\n]", "Since ( p'(1) = 0 ), the root ( x = 1 ) is at least of multiplicity two.", "---", "Step 3: Check Higher-Order Derivatives (Optional for Confirmation)\nTo confirm multiplicity more definitively, we examine successive derivatives. A simple test is whether the first nonzero derivative at ( x = 1 ) corresponds to the smallest power satisfying ( (x - 1)^k ) with ( k \geq 2 ).", "We already have:\n- ( p(1) = 0 )\n- ( p'(1) = 0 )\n- ( p''(x) = \frac{d}{dx}(4x^3 - 12x^2 + 12x - 4) = 12x^2 - 24x + 12 )", "Evaluate ( p''(1) ):", "[\np''(1) = 12(1)^2 - 24(1) + 12 = 12 - 24 + 12 = 0\n]", "Now compute ( p'''(x) ):", "[\np'''(x) = 24x - 24 \quad \Rightarrow \quad p'''(1) = 24(1) - 24 = 0\n]", "And ( p^{(4)}(x) = 24 ), which is nonzero.", "Since the first nonzero derivative at ( x = 1 ) is the fourth (degree 4), the multiplicity of the root ( x = 1 ) is exactly 4, confirming it is a root of multiplicity greater than 1.", "---", "Conclusion:", "We verified that ( x = 1 ) is a root of ( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 ) by satisfying ( p(1) = 0 ) and ( p'(1) = 0 ). The repeated vanishing of the derivative confirms that ( x = 1 ) is a root of multiplicity greater than 1—specifically, a quadruple root.", "This insight is valuable in polynomial analysis, factorization, and root behavior studies.", "---", "Keywords:\nroot multiplicity, verify root multiplicity, polynomial root test, derivative test for multiple roots, ( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 ), ( x = 1 ) root multiplicity, derivative test, zero root analysis.", "Meta Description:\nTo verify if ( x = 1 ) is a root of multiplicity greater than 1 for ( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 ), check if ( p(1) = 0 ) and ( p'(1) = 0 ). Confirming multiplicity helps understand root structure and polynomial factorization."]
