Problem 91 - Algebra 2

Question

Determine whether each polynomial is a perfect square trinomial. $$ x^{2}+18 x+81 $$

Step-by-Step Solution

Answer

Yes, it is a perfect square trinomial.

Step 1: Identify the General Form

A perfect square trinomial has the form \((a+b)^2 = a^2 + 2ab + b^2\). We need to determine if \(x^2 + 18x + 81\) fits this form.

Step 2: Analyze the Given Polynomial

The given polynomial is \(x^2 + 18x + 81\). Here, \(a^2 = x^2\), so \(a = x\) and \(b^2 = 81\), which makes \(b = 9\).

Step 3: Check if Middle Term Matches

For the form \(a^2 + 2ab + b^2\), the middle term should be \(2ab\). Substituting \(a = x\) and \(b = 9\), calculate: \(2ab = 2 \cdot x \cdot 9 = 18x\).

Step 4: Conclusion

The middle term \(2ab = 18x\) matches the middle term of the polynomial \(x^2 + 18x + 81\). This confirms it is a perfect square trinomial.

Key Concepts

Understanding Polynomials

Polynomials are foundational expressions in algebra, composed of variables and constants combined using addition, subtraction, and multiplication.
A single term like a constant or a variable is the simplest form of a polynomial, but they often consist of multiple terms. Key features of polynomials include:

• Degree: The highest power of the variable in the polynomial.
• Terms: Individual components separated by + or - signs, each consisting of a coefficient (a constant) and a variable raised to a power.
• Coefficients: The numeric factor of each term.

In the polynomial expression from our example, we see several important elements.
For example, in the term '18x', 18 is the coefficient, while 'x' is the variable.
The degree is determined by the highest exponent, which is 2, since 'x' is squared.
This degree also indicates that the polynomial is quadratic.

Defining a Trinomial

A trinomial is a specific type of polynomial with exactly three terms.
The terms are usually represented in descending order based on the powers of the variable.
In our example, the trinomial expression is:

• First term: \(x^2\)
• Second term: 18x
• Third term: 81

This arrangement shows that there are three distinct terms, making it a trinomial.
Trinomials are especially common in quadratic equations and often appear in algebra problems.
Their structure allows for specific methods of factoring, such as identifying perfect square trinomials.

Basics of Algebra

Algebra involves working with mathematical symbols and letters to represent numbers in equations and expressions.
It provides a framework to describe relationships between different quantities.
Some fundamental concepts of algebra include:

• Variables: Symbols like x, y, or z that stand in for unknown values.
• Expressions: Combinations of variables, numbers, and operations.
• Equations: Statements declaring the equality of two expressions.

Learning to manipulate these elements is crucial for solving algebraic problems.
For instance, when identifying and confirming a perfect square trinomial, we turn an equation into a familiar algebraic form.
By recognizing patterns, like converting \(x^2 + 18x + 81\) into \((x+9)^2\), we use algebraic techniques to simplify and solve problems related to polynomials.