Answer :

To solve the equation [tex]\(6x^5 - 41 = 151\)[/tex], we start by isolating the power term:

1. Rewrite the Equation:
[tex]\[
6x^5 - 41 = 151
\][/tex]

2. Add 41 to Both Sides to simplify:
[tex]\[
6x^5 = 151 + 41
\][/tex]
[tex]\[
6x^5 = 192
\][/tex]

3. Divide Both Sides by 6 to solve for [tex]\(x^5\)[/tex]:
[tex]\[
x^5 = \frac{192}{6}
\][/tex]
[tex]\[
x^5 = 32
\][/tex]

4. Solve for x by finding the fifth root of 32. The straightforward solution is:
[tex]\[
x = \sqrt[5]{32}
\][/tex]
Since [tex]\(32\)[/tex] is [tex]\(2\)[/tex] raised to the power of [tex]\(5\)[/tex] ([tex]\(2^5 = 32\)[/tex]), this means:
[tex]\[
x = 2
\][/tex]

In addition to this real solution, polynomial equations can have complex solutions as well. Complex solutions arise from the nature of higher-degree polynomial equations which may not all be real numbers. Here are the five solutions, including complex numbers, obtained through solving the polynomial:

- [tex]\(x = 2\)[/tex]
- Four other complex solutions also exist, as polynomials of degree 5 can have up to five solutions in total, including complex ones. These complex solutions are derived mathematically but not elaborated in this explanation.

Therefore, the solutions to the equation [tex]\(6x^5 - 41 = 151\)[/tex] are one real solution [tex]\(x = 2\)[/tex] and four complex numbers.

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Rewritten by : Barada