Answer :
To solve the polynomial equation [tex]\( 5x^4 - 7x^3 - 5x^2 + 5x + 1 = 0 \)[/tex], we need to determine the roots of the equation. The equation is a quartic polynomial, meaning it has the form [tex]\( ax^4 + bx^3 + cx^2 + dx + e = 0 \)[/tex], with [tex]\( a = 5 \)[/tex], [tex]\( b = -7 \)[/tex], [tex]\( c = -5 \)[/tex], [tex]\( d = 5 \)[/tex], and [tex]\( e = 1 \)[/tex].
### Steps to Solve the Quartic Equation
1. Identify the coefficients:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = -5\)[/tex]
- [tex]\(d = 5\)[/tex]
- [tex]\(e = 1\)[/tex]
2. Form the polynomial equation:
[tex]\[ 5x^4 - 7x^3 - 5x^2 + 5x + 1 = 0 \][/tex]
3. Verify that the polynomial is indeed quartic:
Quartic polynomials can have up to 4 roots, which can be real or complex.
4. Use suitable mathematical techniques:
Solving quartic equations directly by hand can be quite complex and typically involves various steps including:
- Depressing the quartic equation (removing the cubic term)
- Solving the resulting simpler polyomial using methods such as substitution or factorization
- In this case, we'll skip over the intermediary steps and directly present the final solutions due to the complexity. Advanced mathematical techniques and symbolic computation tools can derive the exact roots.
### Final Solutions
After performing the detailed and involved steps necessary to solve the quartic equation, the roots of the equation [tex]\(5x^4 - 7x^3 - 5x^2 + 5x + 1 = 0\)[/tex] are given by:
[tex]\[ x_1 = \frac{7}{20} - \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} - \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} - \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} \][/tex]
[tex]\[ x_2 = \frac{7}{20} - \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} + \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} - \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} \][/tex]
[tex]\[ x_3 = \frac{7}{20} - \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} + \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} + \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} \][/tex]
[tex]\[ x_4 = \frac{7}{20} + \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} + \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} + \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} \][/tex]
These solutions involve complex expressions, indicating that the roots may also be complex.
### Steps to Solve the Quartic Equation
1. Identify the coefficients:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = -5\)[/tex]
- [tex]\(d = 5\)[/tex]
- [tex]\(e = 1\)[/tex]
2. Form the polynomial equation:
[tex]\[ 5x^4 - 7x^3 - 5x^2 + 5x + 1 = 0 \][/tex]
3. Verify that the polynomial is indeed quartic:
Quartic polynomials can have up to 4 roots, which can be real or complex.
4. Use suitable mathematical techniques:
Solving quartic equations directly by hand can be quite complex and typically involves various steps including:
- Depressing the quartic equation (removing the cubic term)
- Solving the resulting simpler polyomial using methods such as substitution or factorization
- In this case, we'll skip over the intermediary steps and directly present the final solutions due to the complexity. Advanced mathematical techniques and symbolic computation tools can derive the exact roots.
### Final Solutions
After performing the detailed and involved steps necessary to solve the quartic equation, the roots of the equation [tex]\(5x^4 - 7x^3 - 5x^2 + 5x + 1 = 0\)[/tex] are given by:
[tex]\[ x_1 = \frac{7}{20} - \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} - \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} - \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} \][/tex]
[tex]\[ x_2 = \frac{7}{20} - \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} + \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} - \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} \][/tex]
[tex]\[ x_3 = \frac{7}{20} - \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} + \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} + \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} \][/tex]
[tex]\[ x_4 = \frac{7}{20} + \frac{\sqrt{\frac{347}{150} - 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3} + \frac{43}{500\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}} - \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}}{2} + \frac{\sqrt{\frac{347}{300} + \frac{19}{45(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}} + 2(\frac{4673}{54000} + \sqrt{622119} \cdot i/18000)^{1/3}}}{2} \][/tex]
These solutions involve complex expressions, indicating that the roots may also be complex.
