Questions: Factor the following polynomial. 4 x^2-4 x+1 ([?] x-square)^2

Solution Steps

We start with the polynomial given by:

\[ 4x^2 - 4x + 1 \]

The polynomial can be recognized as a perfect square trinomial. A perfect square trinomial takes the form:

\[ (a - b)^2 = a^2 - 2ab + b^2 \]

In our case, we can identify:

• \(a = 2x\) (since \((2x)^2 = 4x^2\))
• \(b = 1\) (since \(1^2 = 1\))

The middle term \(-4x\) corresponds to \(-2ab\), where \(2ab = 4x\).

Using the identified values of \(a\) and \(b\), we can factor the polynomial as follows:

\[ 4x^2 - 4x + 1 = (2x - 1)^2 \]

Final Answer