Let a be the unique real zero of x^3+x+1 find the simplest possible way to write 18/(a^2+a+1) ^2 as a polynomial in a with rational coefficient

Let \( a \) be the unique real zero of \( x^3 + x + 1 = 0 \). This implies:

\[ a^3 + a + 1 = 0 \]

From this equation, we can isolate \( a^3 \):

\[ a^3 = -a - 1 \]

We need to simplify the expression \(\frac{18}{(a^2 + a + 1)^2}\).

First, let's compute \(a^2 + a + 1\):

Since we already know \( a^3 = -a - 1 \), we also know:

\[ a^3 = -a - 1 \]

This polynomial relationship will be useful to reduce higher powers of \(a\).

Now, let's calculate \((a^2 + a + 1)^2\):

\[ (a^2 + a + 1)^2 = a^4 + 2a^3 + a^2 + 2a^2 + 2a + 1 = a^4 + 2a^3 + 3a^2 + 2a + 1 \]

Next, simplify \(a^4\) and \(2a^3\):

Using \( a^3 = -a - 1 \):

\[ a^4 = a \cdot a^3 = a(-a - 1) = -a^2 - a \]

\[ 2a^3 = 2(-a - 1) = -2a - 2 \]

Substituting these back into the expression:

\[ (a^2 + a + 1)^2 = (-a^2 - a) + (-2a - 2) + 3a^2 + 2a + 1 = -a^2 - a - 2a - 2 + 3a^2 + 2a + 1 \]

Simplifying further:

\[ (a^2 + a + 1)^2 = -a^2 - 3a - 2 + 3a^2 + 2a + 1 \]

\[ = 2a^2 - a - 1 \]

So,

\[ (a^2 + a + 1)^2 = 2a^2 - a - 1 \]

Now we need to find \(\frac{18}{(a^2 + a + 1)^2}\):

\[ \frac{18}{(a^2 + a + 1)^2} = \frac{18}{2a^2 - a - 1} \]

Since \(\frac{18}{2a^2 - a - 1}\) is already in its simplest form, it represents the polynomial in \(a\) with rational coefficients. Hence, the simplest possible way to write it is:

\[ \frac{18}{2a^2 - a - 1} \]

Would you like more details or have any questions? Here are 5 more questions you might be interested in:

1. How do you solve higher-degree polynomials?
2. What is the rational root theorem?
3. How do you factor polynomials?
4. What are the properties of polynomial functions?
5. How do you find the derivative of a polynomial function?

Tip: When working with polynomial equations, always check if there are any obvious factorizations or substitutions that can simplify the problem.

Explore how to simplify the expression 18 divided by the square of (a^2 + a + 1) where 'a' is the unique real zero of the polynomial x^3 + x + 1. Understand the steps involving polynomial simplification and rational coefficient extraction. Suitable for advanced high school students.

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