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To solve the polynomial equation [tex]\( x^4 - x^3 - 14x^2 + 24x + 5 = 0 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that satisfy this equation. Let me take you through the process step-by-step:
1. Identify the Polynomial: We start with the polynomial equation:
[tex]\[ x^4 - x^3 - 14x^2 + 24x + 5 = 0 \][/tex]
2. Substitute a Variable: To solve for the roots, we can use various algebraic methods such as factoring (if straightforward), synthetic division, or more general polynomial root-finding techniques. Here we will solve the equation step-by-step using algebraic methods that lead to finding all the roots of the equation.
3. Decompose and Solve: Given that the polynomial is of degree 4, it generally has four roots. These roots might be real or complex. For a degree 4 polynomial, exact factorization can be challenging, so we use a general approach to find the roots by solving directly.
4. Find the Roots Using Algebraic Methods: This polynomial can be solved algebraically using advanced methods of solving polynomial equations such as the quartic formula, which can be quite complex. The general quartic formula provides the following roots for this equation:
[tex]\[ x_1 = \frac{1}{4} + \frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} - \frac{\sqrt{-\frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} - 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{115}{6}}}{2} \][/tex]
[tex]\[ x_2 = \frac{1}{4} + \frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} + \frac{\sqrt{-\frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} - 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{115}{6}}}{2} \][/tex]
[tex]\[ x_3 = -\frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} + \frac{1}{4} + \frac{\sqrt{-2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} + \frac{115}{6}}}{2} \][/tex]
[tex]\[ x_4 = -\frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} - \frac{\sqrt{-2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} + \frac{115}{6}}}{2} + \frac{1}{4} \][/tex]
5. Conclusion: These expressions represent the four roots of the given polynomial. While they appear complex, they are exact solutions derived from using higher-order algebraic methods to solve quartic equations.
This detailed solution demonstrates various steps from understanding the polynomial to systematically finding the exact roots.
1. Identify the Polynomial: We start with the polynomial equation:
[tex]\[ x^4 - x^3 - 14x^2 + 24x + 5 = 0 \][/tex]
2. Substitute a Variable: To solve for the roots, we can use various algebraic methods such as factoring (if straightforward), synthetic division, or more general polynomial root-finding techniques. Here we will solve the equation step-by-step using algebraic methods that lead to finding all the roots of the equation.
3. Decompose and Solve: Given that the polynomial is of degree 4, it generally has four roots. These roots might be real or complex. For a degree 4 polynomial, exact factorization can be challenging, so we use a general approach to find the roots by solving directly.
4. Find the Roots Using Algebraic Methods: This polynomial can be solved algebraically using advanced methods of solving polynomial equations such as the quartic formula, which can be quite complex. The general quartic formula provides the following roots for this equation:
[tex]\[ x_1 = \frac{1}{4} + \frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} - \frac{\sqrt{-\frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} - 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{115}{6}}}{2} \][/tex]
[tex]\[ x_2 = \frac{1}{4} + \frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} + \frac{\sqrt{-\frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} - 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{115}{6}}}{2} \][/tex]
[tex]\[ x_3 = -\frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} + \frac{1}{4} + \frac{\sqrt{-2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} + \frac{115}{6}}}{2} \][/tex]
[tex]\[ x_4 = -\frac{\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}}{2} - \frac{\sqrt{-2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} - \frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + \frac{135}{4\sqrt{\frac{164}{9\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}}} + 2\left(\frac{\sqrt{99457}}{48} + \frac{12215}{432}\right)^{\frac{1}{3}} + \frac{115}{12}}} + \frac{115}{6}}}{2} + \frac{1}{4} \][/tex]
5. Conclusion: These expressions represent the four roots of the given polynomial. While they appear complex, they are exact solutions derived from using higher-order algebraic methods to solve quartic equations.
This detailed solution demonstrates various steps from understanding the polynomial to systematically finding the exact roots.
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