To factor $x - x^{17}$ completely, follow these steps:

Step 1: Factor out the greatest common factor (GCF)

The GCF in this case is $x$, so we factor out $x$ from the expression:

$x - x^{17} = x(1 - x^{16})$

Step 2: Factor the difference of squares

Notice that $1 - x^{16}$ is a difference of squares, as $1 - x^{16} = (1 - x^8)(1 + x^8)$. So, we factor it as:

$x(1 - x^{16}) = x(1 - x^8)(1 + x^8)$

Step 3: Continue factoring $1 - x^8$

Again, $1 - x^8$ is a difference of squares:

$1 - x^8 = (1 - x^4)(1 + x^4)$

Thus, we now have:

$x(1 - x^8)(1 + x^8) = x(1 - x^4)(1 + x^4)(1 + x^8)$

Step 4: Continue factoring $1 - x^4$

The term $1 - x^4$ is another difference of squares:

$1 - x^4 = (1 - x^2)(1 + x^2)$

Now, substitute this into the expression:

$x(1 - x^4)(1 + x^4)(1 + x^8) = x(1 - x^2)(1 + x^2)(1 + x^4)(1 + x^8)$

Final factored form

The completely factored form of $x - x^{17}$ is:

$x(1 - x^2)(1 + x^2)(1 + x^4)(1 + x^8)$

Would you like more details or have any questions about this factorization?

Related Questions:

1. How do you factor polynomials that involve higher powers?
2. What are the rules for factoring the difference of squares?
3. How can you recognize and factor a perfect square trinomial?
4. What is the process for factoring a sum or difference of cubes?
5. How do you apply factoring techniques to solve polynomial equations?

Tip:

Always check if you can factor further by applying the difference of squares or cubes to simplify higher powers of polynomials.