Question

HIGH DEGREE POLYNOMIAL 9x^{10}-36x^2

Answer 1

To analyze the polynomial \( 9x^{10} - 36x^2 \), we can start by identifying its degree and simplifying it if possible.

1. **Degree of the Polynomial**:
The highest power of \( x \) in the polynomial \( 9x^{10} - 36x^2 \) is \( 10 \). Hence, the degree of this polynomial is \( 10 \).

2. **Factoring the Polynomial**:
We can factor out the greatest common factor (GCF) of the two terms. The GCF of \( 9x^{10} \) and \( -36x^2 \) is \( 9x^2 \).

Factoring out \( 9x^2 \), we get:

\[
9x^2 (x^8 - 4)
\]

3. **Further Factoring**:
The expression \( x^8 - 4 \) can be recognized as a difference of squares:

\[
x^8 - 4 = (x^4 - 2)(x^4 + 2)
\]

The term \( x^4 - 2 \) can also be further factored as a difference of squares:

\[
x^4 - 2 = (x^2 - \sqrt{2})(x^2 + \sqrt{2})
\]

Thus, we can represent the polynomial in its fully factored form:

\[
9x^2 (x^4 - 2)(x^4 + 2) = 9x^2 (x^2 - \sqrt{2})(x^2 + \sqrt{2})(x^4 + 2)
\]

4. **Summary**:
The polynomial \( 9x^{10} - 36x^2 \) is a high-degree polynomial of degree \( 10 \). Its factored form is:

\[
9x^2 (x^2 - \sqrt{2})(x^2 + \sqrt{2})(x^4 + 2)
\]

This expresses the polynomial in terms of its factors. If you want to analyze its roots or other properties, feel free to ask!