How to factorize the expression 4x2y-16xy2+1?

The expression is $4x^2y-16xy^2+1$.
The terms are $4x^2y$, $-16xy^2$, and 1.
To find the greatest common factor (GCF) for all terms, we look at the numerical coefficients and the variables.
The numerical coefficients are 4, $-16$, and 1. The GCF of these numbers is 1.
The variables $x$ and $y$ are present in the first two terms ($4x^2y$ and $-16xy^2$), but not in the third term (1).
Therefore, there is no common factor (other than 1) for all three terms in the expression.

This is a trinomial. We check if it fits common factorization patterns:

• It is not a perfect square trinomial because the last term is 1, but the middle term is not $2 \cdot (\text{first term's square root}) \cdot (\text{last term's square root})$.
• It is not a difference of squares or sum/difference of cubes, as these apply to binomials.
• Factoring by grouping is typically used for four or more terms, or when a common binomial factor can be extracted after grouping. This is not applicable here.

Since there is no common factor for all terms (other than 1) and the expression does not fit any standard factorization patterns, the expression $4x^2y-16xy^2+1$ cannot be factored into simpler polynomials with integer coefficients.

The expression cannot be factored over integers.