Problem 32 - Algebra A Combined Function

Question

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. $$ n^{2}+5 n $$

Step-by-Step Solution

Answer

Add \( \frac{25}{4} \); factor to \( \left(n + \frac{5}{2}\right)^2 \).

Step 1: Identify the Linear Coefficient

The given binomial is \( n^2 + 5n \). The linear coefficient here is 5. This is the coefficient of the term containing \( n \).

Step 2: Divide the Linear Coefficient by 2

We take the linear coefficient (5) and divide it by 2 to get \( \frac{5}{2} \).

Step 3: Square the Result

To complete the square, we square the result from Step 2. \( \left( \frac{5}{2} \right)^2 = \frac{25}{4} \). This is the constant that needs to be added to the binomial.

Step 4: Form the Perfect Square Trinomial

Add \( \frac{25}{4} \) to the original binomial to form the trinomial \( n^2 + 5n + \frac{25}{4} \). This trinomial is a perfect square trinomial.

Step 5: Factor the Trinomial

The trinomial \( n^2 + 5n + \frac{25}{4} \) factors into \( \left(n + \frac{5}{2}\right)^2 \) because we added \( \frac{25}{4} \), which is the square of \( \frac{5}{2} \).

Key Concepts

Understanding Binomials

A binomial is a polynomial expression consisting of exactly two terms. These terms are typically linked together by either addition or subtraction. In our example, the binomial is \( n^2 + 5n \). The term "binomial" originates from "bi-", meaning two, and "-nomial", relating to terms. This concept makes binomials one of the simpler forms of expressions within algebra. Binomials can take various forms such as:

• Two variable expressions: \( ax + b \)
• Squared terms: \( x^2 + 2xy \)

Each binomial has its own identity and set of possible transformations. In our task, we need to convert this binomial into a trinomial that is a perfect square.

The Role of the Linear Coefficient

The linear coefficient is an essential component of any polynomial and specifically a binomial. It is the number that scales the variable without an exponent attached to it. In the given binomial \( n^2 + 5n \), 5 is the linear coefficient. It is crucial because it helps determine how we can transform a binomial into a perfect square trinomial. To find the necessary constant for completing the square, the linear coefficient is divided by 2. In our case:

• Linear Coefficient: 5
• Divide by 2: \( \frac{5}{2} \)

This step is foundational because this calculation ultimately influences the entire transformation process.

The Importance of Squaring

Squaring a number means multiplying it by itself. This operation is crucial in the process of converting a binomial into a perfect square trinomial. After halving the linear coefficient in the binomial, we next square that result to find the additional constant we need.In our example:

• Half of the linear coefficient: \( \frac{5}{2} \)
• Squared: \( \left(\frac{5}{2}\right)^2 = \frac{25}{4} \)

This squared value, \( \frac{25}{4} \), is added to the original binomial. By doing this, we ensure the created trinomial represents a perfect square.

Recognizing a Trinomial

A trinomial, as the name suggests, is a polynomial consisting of three distinct terms. In our example, the perfect square trinomial is \( n^2 + 5n + \frac{25}{4} \). When forming a perfect square trinomial, each of the terms has its specific role, aligning often in the format \( a^2 + 2ab + b^2 \).Trinomials are significant because:

• They allow expression in a single squared binomial: \( (a + b)^2 \)
• They simplify the solving of equations and enhance understanding of polynomial behavior.

By transforming our original binomial into this trinomial, we can now say it factors neatly into \( \left(n + \frac{5}{2}\right)^2 \), illustrating the elegance and utility of algebraic structures.