To factor the given polynomial \( 64x^4 - 16x^3 + 17x^2 - 2x + 1 \), let's attempt to factor it step by step.

### 1. Identify Possible Rational Roots:
By the Rational Root Theorem, possible rational roots are the factors of the constant term (1) divided by the factors of the leading coefficient (64). So the possible rational roots could be \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{1}{16}, \pm \frac{1}{32}, \pm \frac{1}{64} \).

### 2. Testing Rational Roots:
We can test these values using synthetic division or substitution to see if they are roots. If any value is a root, then it factors the polynomial.

Let me test these possible roots for you.None of the tested rational roots yield a value of zero, which means none of these possible roots are actual roots of the polynomial \( 64x^4 - 16x^3 + 17x^2 - 2x + 1 \). Since the polynomial does not factor nicely over the rationals, we will need to explore other methods such as numerical methods, factoring over complex numbers, or recognizing if the polynomial might be irreducible.

Would you like to continue with a numerical approach, attempt complex factorization, or explore other methods? Please specify your preference.

Learn how to factor the polynomial 64x^4 - 16x^3 + 17x^2 - 2x + 1 with a detailed step-by-step solution. This problem involves understanding polynomials and applying the Rational Root Theorem to identify potential roots. Suitable for advanced high school students (Grades 10-12).

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