Question

pregunta 1-2
which expression is equivalent to ( x^4 - 2x^3 + x^2 )?
( circ x^2(x - 1)^2 )
( circ x^2(x + 1)^2 )
( circ x^2(x + 1)(x - 1) )
( circ x^2(x^2 + 1)(x - 1) )

Explanation:

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

The GCF of \(x^4\), \(-2x^3\), and \(x^2\) is \(x^2\). So we factor out \(x^2\) from the expression:
\(x^4 - 2x^3 + x^2 = x^2(x^2 - 2x + 1)\)

Step 2: Factor the quadratic expression

Now we factor the quadratic \(x^2 - 2x + 1\). Notice that it is a perfect square trinomial, since \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 1\), so:
\(x^2 - 2x + 1 = (x - 1)^2\)

Step 3: Combine the factors

Putting it all together, we have:
\(x^2(x^2 - 2x + 1) = x^2(x - 1)^2\)

Answer:

\(x^2(x - 1)^2\) (corresponding to the first option: \(x^2(x - 1)^2\))