Problem 12 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ x^{2}+18 x+81 $$

Step-by-Step Solution

Answer

The trinomial factors as \( (x + 9)^2 \).

Step 1: Identify the Form

The given trinomial is in the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 18 \), and \( c = 81\).

Step 2: Check for a Perfect Square

A perfect square trinomial is of the form \( (x + d)^2 \). Here, we calculate if \((b/2)^2 = c\). Calculate \((18/2)^2 = 9^2 = 81\). This matches \(c\), so the trinomial is a perfect square.

Step 3: Write the Expression as a Square

Since the trinomial is a perfect square, it can be expressed as \( (x + 9)^2 \). Both \((x + 9)(x + 9)\) yield the same trinomial when expanded.

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are a special type of trinomial in algebra that can be written in the square of a binomial form. Recognizing and working with perfect square trinomials is an essential skill in algebraic manipulation and factoring. Typically, these trinomials appear in the form of:

• \((x + d)^2 = x^2 + 2dx + d^2\)
• \((x - d)^2 = x^2 - 2dx + d^2\)

The challenge lies in identifying them within an algebraic expression, which involves checking if the middle term \(2dx\) aligns correctly with \(d\). A quick check involves computing \((b/2)^2\) and seeing if it equals \(c\), where \(b\) and \(c\) are the coefficients in \(ax^2 + bx + c\).

In our example, \(x^2 + 18x + 81\), breaking it down reveals that \((18/2)^2 = 81\), which confirms it's a perfect square. Thus, the original expression simplifies neatly into \((x + 9)^2\). This method saves time and effort, making calculation simpler and more efficient.

Factoring Techniques

Factoring is an important technique in simplifying algebraic expressions, solving equations, and understanding polynomial structures. Factoring often involves breaking down a composite expression into simpler factors. For trinomials, it particularly helps when reducing them to a product of binomials or perfect square factors.

When dealing with trinomials, the first step is to identify if they are perfect squares or can be factored straightforwardly using known algebraic identities. There are several methods:

• Trial and error
• Using the quadratic formula
• Recognizing certain patterns like perfect square trinomials

Each technique can be useful depending on the specific trinomial you are working with.

In the original exercise example, checking for a perfect square simplifies the process immensely. Upon confirming it's a perfect square trinomial, it can efficiently be expressed as \((x + 9)^2\). Thus, applying appropriate techniques ensures quicker and more accurate solutions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In algebra, understanding how to manipulate these expressions is foundational. Expressions can range from simple forms, like \(x + 2\), to more complex ones, such as trinomials and polynomials.

Working with these forms often involves:

• Performing operations like addition, subtraction, and multiplication
• Factoring or expanding expressions
• Simplifying terms

In the realm of polynomial expressions, trinomials are commonly encountered in both homework and exams. Being able to factor these into irreducible components not only aids in solving algebraic equations but also enhances general problem-solving skills in mathematics.

In the given exercise \(x^2 + 18x + 81\), recognizing that this expression can be reduced to \((x + 9)^2\) through factoring underscores the entire purpose of mastering algebraic manipulation—simplification. These skills are not just vital for academic exercises, but also for real-world problem solving and mathematical reasoning.