We are tasked with finding the range of values of $a^2$ such that the fourth-degree polynomial $x^4 - 2ax^2 + x + a^2 - a = 0$ has all real roots.

Let's break this problem down:

The given polynomial is: $x^4 - 2ax^2 + x + a^2 - a = 0$ To find the range of $a$ that ensures all the roots are real, we'll need to analyze the behavior of the roots of this polynomial. Fourth-degree polynomials can have various combinations of real and complex roots, but for all roots to be real, we have to ensure the discriminant is non-negative (this is one approach, though working with higher-degree polynomials is more complex than quadratics).

Step 1: Analyze the polynomial

The polynomial $x^4 - 2ax^2 + x + a^2 - a = 0$ can be seen as a combination of terms involving both $x^4$ and $x^2$. Let’s examine this step-by-step by considering potential substitutions and analyzing its discriminant.

We can make the substitution $y = x^2$, which transforms the fourth-degree polynomial into a quadratic in terms of $y$.

1. $y^2 = x^4$
2. The term $-2ax^2$ becomes $-2ay$.

Thus, the polynomial becomes: $y^2 - 2ay + x + a^2 - a = 0$ Now, we need to analyze the roots of this transformed equation.

Step 2: Consider discriminant conditions

The discriminant of a polynomial is a key indicator of the nature of its roots (real vs. complex). For all real roots, the discriminant of the polynomial must be non-negative.

To compute the discriminant, we would typically differentiate the function to find critical points and study the nature of these points, but it is somewhat complex due to the non-standard terms.

Step 3: Approach by graphing or solving numerically

To find the exact range of $a$ that ensures all real roots, the most practical approach is to either:

• Graph the polynomial for different values of $a$ and observe when all the roots become real.
• Use numerical methods to compute the roots for various values of $a$.

Without more explicit constraints, solving this analytically may be difficult, but based on typical behavior of fourth-degree polynomials, this type of problem often has specific ranges of $a$ (which might involve numerical bounds or algebraic constraints).

Would you like to explore solving this numerically or graphically?

Here are 8 questions to further explore this:

1. Would you like to explore a graphical method to analyze the roots of the polynomial?
2. Should we substitute specific values of $a$ and check the nature of the roots?
3. Would calculating the discriminant for a specific form help in this case?
4. Are you interested in the role of symmetry in the polynomial’s roots?
5. Would factoring the polynomial into simpler components be useful here?
6. Can we use numerical root-finding algorithms for this specific polynomial?
7. Are there any constraints on the values of $a$ from a real-world context?
8. Should we apply a specific numerical method, like Newton’s method, to find the roots?

Tip: For higher-degree polynomials, numerical or graphical methods are often very helpful in finding the exact nature of the roots, especially when analytical solutions are complex.