Question

Factor the perfect squares. $$ x^{2}-4 x+4 $$

Short answer

\((x - 2)^2\)

Step-by-step solution

Identify the Perfect Square Trinomial

Recognize that the expression is a perfect square trinomial of the form \[a^2 - 2ab + b^2\] where \(a = x\) and \(b = 2\).

Rewrite the Expression

Rewrite the trinomial in the form \((a - b)^2\). In this case, \(x^2 - 4x + 4\) can be written as \((x - 2)^2\).

Verify the Factored Form

Expand \((x - 2)^2\) to ensure it equals the original expression. \((x - 2)^2 = x^2 - 4x + 4\), which matches the given polynomial.

Key concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of polynomial. It is in the form a^2 or -2ab + b^2 This means the trinomial can be factored into a binomial squared. For example, consider x^2 - 4x + 4 . This follows the pattern because = 2x and b^2 = 4. So, the trinomial can be rewritten and factored as (x - 2)^2 . Detecting a perfect square understands the signatures and patterns. If you see squares of numbers and twice the product of two numbers, likely, you've encountered a perfect square trinomial.

Factoring Technique

Factoring a trinomial involves several steps. Initially, we need to identify the structure. For a perfect square trinomial, it matches a^2 − 2ab + b^2 or a^2 + 2ab + b^2 . Once identified, rewrite it in a squared binomial form. Taking x^2 - 4x + 4 as an example: Step 1: Identify the form: Here, x =a, and the product 2ab = 4x, thus b = 2. Step 2: Rewrite the expression using (a -b)^2 . Hence, (x-2)^2 . Step 3: Verify: Expand ( x-2)^2 to ensure it equals the original trinomial. Verifying ensures we have not made a mistake. Thus practicing lays the foundation of mastering any factoring technique.

Polynomial

A polynomial is a sum of terms, each term including a variable raised to a power and a coefficient. They come in different forms such as monomials, binomials, trinomials, etc., based on the number of terms. The standard form of a polynomial arranges terms in descending powers of the variable. For example, x^2 - 4x + 4 is a polynomial where: - x^2 is a quadratic term - 4x is a linear term - 4 is a constant term Trinomials have three terms and can often be factored into binomials. Recognizing each term's contributions to the polynomial's overall structure is key to solving and simplifying them. Studying polynomials helps tackle more complex algebraic expressions with confidence.