Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Factor the perfect square trinomial.\(x^{2}-4 x+4\)

Step-by-Step Solution

Answer

The factored form of the given perfect square trinomial \(x^{2} - 4x + 4\) is \((x-2)^{2}\).

Step 1: Identify the Form

A perfect square trinomial is a trinomial of the form \((a - b)^{2}\) or \((a + b)^{2}\). In each case, the trinomial can be broken down to \(a^{2} - 2ab + b^{2}\) or \(a^{2} + 2ab + b^{2}\) respectively. In this exercise, the trinomial is of the form \(a^{2} - 2ab + b^{2}\), because it includes negative term.

Step 2: Identify 'a' and 'b'

From the standard form \(a^{2} - 2ab + b^{2}\), compare this with the given expression \(x^{2} - 4x + 4\). Here, \(a^2 = x^2\), so \(a = x\). And \(b^2 = 4\), hence \(b = 2\).

Step 3: Write in Factored Form

To write the trinomial in factored form, use the equation \((a-b)^{2}\). Replace 'a' with 'x' and 'b' with '2'. The factored form is \((x-2)^{2}\).

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is one of the unique and predictable polynomial types you will encounter in algebra. It follows a specific pattern based on the expression

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

This structured form allows us to easily factor the trinomial back into a binomial squared. In our given example, the trinomial \(x^{2} - 4x + 4\) fits the pattern of \(a^{2} - 2ab + b^{2}\). This is because:

• The first term \(x^{2}\) is a perfect square of \(x\).
• The last term \(+4\) is also a perfect square, which is \(2^{2}\). This reveals that the middle term \(-4x\) must be \(-2 imes x imes 2\).

Once these components align, we confirm it as a perfect square trinomial, enabling us to express it in compact factored form.

Discovering the Factored Form

Converting a perfect square trinomial into its factored form is straightforward if you recognize the trinomial pattern. This process translates a longer polynomial into a simpler, more condensed expression.

• First, identify ‘a’ and ‘b’ from the trinomial’s structure: here, \(a = x\) and \(b = 2\).
• Then, employ the pattern: since the given trinomial is \(x^{2} - 4x + 4\), it fits \((a - b)^{2}\).
• Thus, the factored form is \((x - 2)^{2}\).

This expression \((x - 2)(x - 2)\) is equivalent to the original trinomial. Factoring simplifies solving, graphing, or analyzing polynomials by reducing them to a product of binomials.

Exploring Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using arithmetic operations such as addition, subtraction, multiplication, and division. These expressions form the backbone of algebra, enabling us to represent and solve real-world problems mathematically. In this problem,

• The expression \(x^{2} - 4x + 4\) is an algebraic expression involved in a process called factoring.
• In general, expressions can include constants like \(4\) or variables like \(x\), which can represent various numbers.

Understanding how to manipulate these expressions is crucial in algebra.

Through factoring, we can decompose complex expressions into simpler parts, making it easier to interpret and solve algebra problems. Whether it's evaluating them, simplifying them, or preparing them for equations, mastery over algebraic expressions is fundamental to success in mathematics.