To find all the solutions of the polynomial equation [tex]\(x^4 - 4x^3 + x^2 + 8x - 6 = 0\)[/tex], we need to identify the roots of the polynomial.
The correct solutions to the given polynomial equation are:
[tex]\[
x = 1, x = 3, x = -\sqrt{2}, x = \sqrt{2}
\][/tex]
Here’s a detailed step-by-step explanation for solving the polynomial equation:
1. Understand the polynomial: We start with the equation:
[tex]\[
x^4 - 4x^3 + x^2 + 8x - 6 = 0
\][/tex]
A polynomial of degree 4 suggests there could be up to 4 roots, some of which might be real, and some might be complex.
2. Identify the solutions: The roots of the polynomial [tex]\(x^4 - 4x^3 + x^2 + 8x - 6 = 0\)[/tex] are:
[tex]\[
x = 1, x = 3, x = -\sqrt{2}, x = \sqrt{2}
\][/tex]
3. Check possible root options: Among the provided multiple choices, the correct set of roots matches only one of the given options.
4. Verify the correct answer: We see that the correct answer must include [tex]\( \pm \sqrt{2}, 1, \)[/tex] and [tex]\( 3 \)[/tex]. Comparing with the options provided:
- [tex]\(x= \pm 2, x =1, x =-3\)[/tex]
- [tex]\(x= \pm 1, x= \pm 2, x= \pm 3\)[/tex]
- [tex]\(x= \pm i, x =-3, x =2\)[/tex]
- [tex]\(x= \pm \sqrt{2}, x =1, x =3\)[/tex]
The correct set of answers is:
[tex]\[
x= \pm \sqrt{2}, x =1, x =3
\][/tex]
Therefore, the solutions to the polynomial equation [tex]\(x^4 - 4x^3 + x^2 + 8x - 6 = 0\)[/tex] are [tex]\( x = \pm \sqrt{2}, x = 1, \)[/tex] and [tex]\( x = 3\)[/tex].
