factorise x^2 - 2x + 1 and elaborate on the steps
user
Sure, I'd be happy to help you factorize the polynomial `x^2 - 2x + 1`. Here are the steps:

1. If you move a constant term through a subtraction sign and attach it to the variable, the two signs change. That is, if `A` is a real number, the following two expressions are equivalent:

   `Ax - B = Ax + (-B)`, and `(-B) = -1 * B`.

   So, `x^2 - 2x + 1 = x^2 + (-2)x + 1 = x(x + 2) + 1`.

2. Now observe that `x + 2` is a common factor of `x^2` and the second term `-2x`.

   You have found the greatest common factor, GCF, of `x^2` and `2x` by extracting the unit scalar `x`.

   But since `1` must also divide both the same unit scalar, so the GCF is `x(x + 2) = x^2 + 2x + x = x(x + 2)`.

3. Next, take out the common factor `x(x + 2)`, which is parentheses, and rewrite the polynomial as `x(x + 2) = 1 * x * (x + 2)`.

4. As a result, it is written as the multiplication of x and (x+2), covered with *-helps, the result is the same.

In conclusion, `x^2 - 2x + 1 = x*(x+2)`.

Hope it helps!
assistant
factorise x^2 - 2x + 1 and elaborate on the steps
user
Sure, I'd be happy to help with that! The expression you've given is a quadratic, and it can be factored using the perfect square trinomial pattern. Here are the steps:

1. **Identify the pattern**: The given quadratic expression is in the form of a perfect square trinomial, which is a square of a binomial. The general forms of perfect square trinomials are:
   - (a + b)² = a² + 2ab + b²
   - (a - b)² = a² - 2ab + b²

2. **Match the terms**: In the given expression x² - 2x + 1, we can see that:
   - x² is the square of x, so it corresponds to a² in the general forms.
   - 1 is the square of 1, so it corresponds to b² in the general forms.
   - -2x corresponds to the middle term, -2ab or 2ab, in the general forms. Since it's negative, we know we're dealing with the (a - b)² form.

3. **Factor the expression**: Since we've identified that the expression is a perfect square trinomial of the form (a - b)², we can factor it as (x - 1)².

So, x² - 2x + 1 factors to (x - 1)². This means that the original expression is the square of the binomial (x - 1).
assistant