Answer :
To solve the polynomial equation [tex]\(2x^4 + 3x^3 - 7x^2 - 8x + 6 = 0\)[/tex], you would typically follow these steps:
### Step 1: Identify the Polynomial
The polynomial in question is:
[tex]\[ 2x^4 + 3x^3 - 7x^2 - 8x + 6 \][/tex]
### Step 2: Check for Possible Rational Roots
Use the Rational Root Theorem, which states that any rational solution of the polynomial equation [tex]\(P(x) = 0\)[/tex] is a fraction [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] is a factor of the constant term (in this case, 6), and [tex]\(q\)[/tex] is a factor of the leading coefficient (in this case, 2). The possible rational roots are therefore:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \][/tex]
Testing these possible roots can be labor-intensive and might not yield any solutions or only some of the solutions.
### Step 3: Using Numerical or Symbolic Methods
Since we have a quartic (degree 4) polynomial, finding roots might require symbolic computation or numerical methods to find exact or approximate values.
### Step 4: Solving the Equation for Exact Roots
Using more advanced algebraic methods, you may find that the solutions are complex and involve nested square roots and possibly the use of the quadratic formula for higher degrees.
Given the complexity, we find the roots to be:
[tex]\[ -\frac{3}{8} + \sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} + \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} \][/tex]
[tex]\[ -\sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} - \frac{3}{8} - \sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} - \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} \][/tex]
[tex]\[ -\sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} - \frac{3}{8} + \sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} - \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} \][/tex]
[tex]\[ -\sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} - \frac{3}{8} + \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} \][/tex]
These roots are complex and do not easily simplify further in typical algebraic methods.
### Final Note:
The solutions provided are exact and involve intricate expressions. In practice, one may use numerical approximations for simplicity unless exact solutions are required.
### Step 1: Identify the Polynomial
The polynomial in question is:
[tex]\[ 2x^4 + 3x^3 - 7x^2 - 8x + 6 \][/tex]
### Step 2: Check for Possible Rational Roots
Use the Rational Root Theorem, which states that any rational solution of the polynomial equation [tex]\(P(x) = 0\)[/tex] is a fraction [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] is a factor of the constant term (in this case, 6), and [tex]\(q\)[/tex] is a factor of the leading coefficient (in this case, 2). The possible rational roots are therefore:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \][/tex]
Testing these possible roots can be labor-intensive and might not yield any solutions or only some of the solutions.
### Step 3: Using Numerical or Symbolic Methods
Since we have a quartic (degree 4) polynomial, finding roots might require symbolic computation or numerical methods to find exact or approximate values.
### Step 4: Solving the Equation for Exact Roots
Using more advanced algebraic methods, you may find that the solutions are complex and involve nested square roots and possibly the use of the quadratic formula for higher degrees.
Given the complexity, we find the roots to be:
[tex]\[ -\frac{3}{8} + \sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} + \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} \][/tex]
[tex]\[ -\sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} - \frac{3}{8} - \sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} - \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} \][/tex]
[tex]\[ -\sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} - \frac{3}{8} + \sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} - \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} \][/tex]
[tex]\[ -\sqrt{-2\left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} - \frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + \frac{61}{32 \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}} + \frac{139}{24}}}{2} - \frac{3}{8} + \sqrt{\frac{265}{72 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}}} + 2 \left(\sqrt{\frac{65811}{576} + \frac{2191}{864}}\right)^{\frac{1}{3}} + \frac{139}{48}}}{2} \][/tex]
These roots are complex and do not easily simplify further in typical algebraic methods.
### Final Note:
The solutions provided are exact and involve intricate expressions. In practice, one may use numerical approximations for simplicity unless exact solutions are required.
