If \( x^3 - 7x^2 + 14x - 8 = 0 \) has a root at \( x = 1 \), find the other roots.
["Title: Find All Roots of the Cubic Equation ( x^3 - 7x^2 + 14x - 8 = 0 ) Given One Root", "---", "Introduction\nSolving cubic equations can seem challenging, but when one root is known, the equation simplifies greatly. This SEO-optimized article explores how to find the remaining roots of the cubic equation\n[\nx^3 - 7x^2 + 14x - 8 = 0\n]\ngiven that ( x = 1 ) is a root. We’ll use polynomial division and factorization techniques to uncover all three roots efficiently.", "---", "Step 1: Confirming ( x = 1 ) as a Root\nWe verify that ( x = 1 ) satisfies the equation:\n[\n1^3 - 7(1)^2 + 14(1) - 8 = 1 - 7 + 14 - 8 = 0\n]\nSince the result is zero, ( x = 1 ) is indeed a root.", "---", "Step 2: Factor the Cubic Polynomial Using Synthetic Division\nTo eliminate ( x - 1 ) from the cubic polynomial, we perform synthetic division.", "We divide ( x^3 - 7x^2 + 14x - 8 ) by ( x - 1 ):", "<br/>\n1 | 1 -7 14 -8<br/>\n | 1 -6 8</p>\n<hr/>\n<pre><code> 1 -6 8 0\n</code></pre>\n<p>", "The quotient is ( x^2 - 6x + 8 ), and the remainder is 0, confirming divisibility.", "Thus,\n[\nx^3 - 7x^2 + 14x - 8 = (x - 1)(x^2 - 6x + 8)\n]", "---", "Step 3: Factor the Quadratic\nNow solve the quadratic equation:\n[\nx^2 - 6x + 8 = 0\n]\nFactor the quadratic:\n[\nx^2 - 6x + 8 = (x - 2)(x - 4)\n]\n(Check: ( (x - 2)(x - 4) = x^2 - 6x + 8 ))", "---", "Step 4: Find All Roots\nFrom the factorization:\n[\n(x - 1)(x - 2)(x - 4) = 0\n]\nSet each factor equal to zero:\n[\nx - 1 = 0 \Rightarrow x = 1\n]\n[\nx - 2 = 0 \Rightarrow x = 2\n]\n[\nx - 4 = 0 \Rightarrow x = 4\n]", "---", "Conclusion\nGiven that ( x = 1 ) is a root of the equation ( x^3 - 7x^2 + 14x - 8 = 0 ), the other two roots are:\n[\n\boxed{x = 2 \quad \ ext{and} \quad x = 4}\n]\nThis gives the full set of roots: ( x = 1, 2, 4 ).", "---", "SEO Keywords: \nCubicEquationRoots, #SolveCubicEquation, #FactorCubicPolynomial, #FindRoots, #PolynomialFactorization, #RootsOfPolynomial, #AlgebraSolve, #MathTips, #ElementaryAlgebra", "Meta Description:\nGiven ( x^3 - 7x^2 + 14x - 8 = 0 ) has a root at ( x = 1 ), this article explains how to find the other roots using synthetic division and quadratic factorization. Learn step-by-step.", "---", "Boost Your Search Potential:\n- Target long-tail queries like “how to factor cubic equations,”\n- Include FAQs such as “what are the roots of ( x^3 - 7x^2 + 14x - 8 = 0 )?”\n- Link to more detailed algebraic guides and equations practice problems.", "---", "Final Thoughts:\nIdentifying one root unlocks the entire solution. This method applies not only to quadratics but extends smoothly to cubics—essential for math students, educators, and anyone mastering algebra.", "---", "Key Takeaway:\nWhen given a root of a polynomial, use polynomial division to reduce the degree, then solve the lower-degree factors. Fast, accurate, and scalable—this technique ensures you never miss a root again.", "---", "Related Reads:\n- How to Factor Higher-Degree Polynomials\n- Solving Quadratic Equations by Factoring\n- Understanding Cubic Polynomials: Real Roots and Factorization\n- Step-by-Step Guide to Solving Cubic Equations", "---", "Keywords for technical SEO:\n( x^3 - 7x^2 + 14x - 8 = 0 ), roots of cubic, factor cubic, synthetic division, polynomial root finding, cubic factors, algebraic equations, math explanation, polynomial factorization, elementary algebra."]
