Question

Factor the perfect squares. $$ x^{2}-2 x+1 $$

Short answer

The expression factors to \[ (x - 1)^2 \].

Step-by-step solution

- Identify the form

Recognize that the given expression, \(x^{2}-2x+1\), is a quadratic expression that can potentially be factored into a perfect square. A perfect square takes the form \( (a - b)^{2} \).

- Identify \(a\) and \(b\)

To factor the expression into a perfect square, identify the values of \(a\) and \(b\). The coefficient of \(x^{2}\) is 1, so \(a = x\). The constant term is 1, implying that \(b = 1\). Verify by checking the middle term: \(2ab = 2 \times x \times 1 = 2x\). This matches the original expression.

- Write as a squared binomial

Since \(a = x\) and \(b = 1\), we can write the given expression as a squared binomial. Therefore, \[ x^{2} - 2x + 1 = (x - 1)^2 \].

Key concepts

perfect squares

Perfect squares are expressions that can be written as the square of a binomial. In the exercise, the quadratic expression is identified as a perfect square. For a quadratic expression to be a perfect square trinomial, it must fit the pattern \(a^2 \pm 2ab + b^2\). This means that both the first and last terms of the quadratic must be perfect squares themselves, and the middle term must be twice the product of the square roots of the first and last terms.

quadratic expressions

Quadratic expressions are polynomial expressions of degree two. The general form of a quadratic expression is \(ax^2 + bx + c\), where \(a, b,\) and \c\ are constants, and \(x\) represents the variable. Factoring quadratics often involves recognizing patterns such as perfect squares or using methods like factoring by grouping, completing the square, or the quadratic formula. Understanding these methods helps simplify solving quadratic equations, like turning them into a product of two binomials.

squared binomial

A squared binomial is an expression that results from squaring a binomial. A binomial is a polynomial with two terms, and squaring it involves multiplying it by itself. For example, squaring the binomial \( (a - b) \) results in the expression \(a^2 - 2ab + b^2\), which is a perfect square trinomial. In the provided exercise, \( (x - 1)^2\) expands to \(x^2 - 2x + 1\). Recognizing and writing expressions as squared binomials simplifies the factorization process.