Question

Is any real number exactly 1 less than its fourth power?
a) Yes
b) No

Answer 1

Yes, there is a real number that is exactly 1 less than its fourth power. This is found by solving the equation x^4 - x - 1 = 0, which has at least one real solution.

Step-by-step explanation:

The question asks whether there is a real number that is exactly 1 less than its fourth power. To find the answer, we need to set up an equation where a real number x equals its fourth power minus 1, which is x^4 - x - 1 = 0. The powers of a number are part of basic algebra, often discussed in high school mathematics.

To solve this equation, we usually look for rational roots using the Rational Root Theorem, or we can graph the function and look for where it crosses the x-axis. This equation does not factor nicely, so we would need to resort to numerical methods or graphing to find a solution. Through graphing or using advanced solving techniques, we can find that there is indeed at least one real solution to this equation.

Answer 2

No real number is exactly 1 less than its fourth power (Option B).

Step-by-step explanation:

Assume there exists a real number "x" such that x^4 = x - 1.

Rearrange the equation to x^4 - x + 1 = 0.

This is a quartic equation, but it doesn't have real roots because the expression x^4 - x + 1 is always positive for real x.

By Descartes' Rule of Signs, there are no sign changes, indicating no positive real roots.

Additionally, the quartic equation's discriminant is negative, confirming no real roots.

Therefore, no real number satisfies x^4 = x - 1, and the answer is No (Option B).