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The ***Greatest Common Factor (GCF)*** is the largest factor shared by all terms in a polynomial. In our example, the polynomial is: $$ x^{13} - 6 x^{7} + 9 x $$. To find the GCF, identify any common term or number shared across all parts of the polynomial. Notice that each term in our polynomial contains at least one factor of ***x***. Thus, the GCF is ***x***. This will be our starting point when factoring the polynomial.

Recognizing quadratic forms is crucial when dealing with higher-degree polynomials. A quadratic form looks like ***ax^2 + bx + c***. In this case, after factoring out the GCF, we have: $$x(x^{12} - 6x^6 + 9)$$ Here, the expression inside the parentheses, ***x^{12} - 6x^6 + 9***, can be seen as a quadratic form if we set: ***y = x^6***. Thus, ***x^{12} = (x^6)^2 = y^2***, making our expression look like: ***y^2 - 6y + 9***. Recognizing this form helps in the next steps of factoring.

Factoring quadratic expressions follows a straightforward approach once we recognize the quadratic form. From our redefined polynomial: $$ y^2 - 6y + 9 $$ We see that it is a simple quadratic. To factor it, find two numbers that multiply to the constant term (9) and add up to the linear term coefficient (-6). These two numbers are both ***-3***. Now, factor the quadratic expression as: $$ (y - 3)(y - 3) = (y - 3)^2 $$ Finally, substitute back the original variable, which we had set as ***y = x^6***, resulting in: $$ (x^6 - 3)^2 $$. When we include the previously factored out GCF, we have the fully factored polynomial: $$ x(x^6 - 3)^2 $$. This method ensures we correctly break down and simplify the polynomial.