Problem 49 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}-2 x+1$$

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(x^{2} - 2x + 1\) is \((x - 1)^2\).

Step 1: Identify the terms

First, identify the terms from the trinomial equation \(x^{2} - 2x + 1 = 0\). Here, 'a' is x (from \(a^{2}\)), 'b' is 1 (from -2ab and \(b^{2}\)) and the constant term is -2 (from -2ab).

Step 2: Check if it's a perfect square trinomial

Now verify if the trinomial is a perfect square. A trinomial is a perfect square when the square of half of the coefficient of the term containing 'x' is equal to the constant term. Here, half of the coefficient of 'x' is -1 (\(-2/2\)), and its square is 1 (\((-1)^2\)), which is equal to the constant term in the trinomial. Therefore, the trinomial \(x^{2} - 2x + 1\) is a perfect square trinomial.

Step 3: Factor the trinomial

Perfect square trinomials can be expressed in the form of \((a - b)^2\), or \((a+b)^2\) depending on whether the coefficient of 'b' in the trinomial is negative or positive. We identified 'a' as x and 'b' as 1, and since the coefficient of 'b' is negative in our case, we use the form \((a - b)^2\). Substituting the values, the factored form of the given trinomial is \((x - 1)^2\).

Key Concepts

Understanding Perfect Square Trinomials

Perfect square trinomials are a specific form of polynomials that you can easily factor because they follow a predictable pattern. A trinomial is a perfect square if it can be written as the square of a binomial. This means it takes the form

• \((a + b)^2 = a^2 + 2ab + b^2\)
• or \((a - b)^2 = a^2 - 2ab + b^2\)

You recognize a perfect square trinomial if the first and last terms are perfect squares themselves, and the middle term is twice the product of their square roots.
For example, the trinomial \(x^2 - 2x + 1\) is a perfect square because:

• The first term \(x^2\) is the square of \(x\).
• The last term 1 is the square of 1.
• The middle term \(-2x\) equals \(-2 \times x \times 1\), which confirms the pattern.

Factoring it gives us \((x-1)^2\), showing that the trinomial is indeed a perfect square.

Identifying a Polynomial Prime

A polynomial is considered prime when it cannot be factored into simpler polynomial expressions with whole number coefficients. In other words, if there are no two binomials that multiply to give the polynomial, it remains a prime polynomial.
When checking if a polynomial is prime, ensure no factors or special patterns apply, such as perfect square trinomials or difference of squares.
For instance, had \(x^2 - 2x + 1\) not fit the form of a perfect square trinomial, and no other factorization was possible, we would classify it as a prime polynomial. Recognizing prime polynomials helps determine if simplification or alternative problem-solving strategies are needed.

Steps to Factor Trinomials

Factoring trinomials involves writing them as a product of two binomial expressions. Understanding and applying the process efficiently can simplify many algebraic operations. To factor a trinomial like \(ax^2 + bx + c\), follow these key steps:

• Identify the 'a', 'b', and 'c' coefficients from the trinomial.
• Check if it's a perfect square trinomial first, as they follow the predictable patterns of \((a \pm b)^2\).
• If not a perfect square, find two numbers that multiply to ac (product of the first and last coefficient) and add up to b.
• Use these numbers to break the middle term and factor by grouping.
• Check your work by multiplying the binomials to see if they return the original trinomial.

For \(x^2 - 2x + 1\), the process was simplified as it clearly fit the formula of a perfect square trinomial, enabling quickly factorization as \((x-1)^2\).