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Chapter 11: Problem 32

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

Short Answer

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

Step by step solution

01

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 \( n \).
02

Divide the Linear Coefficient by 2

We take the linear coefficient (5) and divide it by 2 to get \( \frac{5}{2} \).
03

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.
04

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.
05

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 \( \frac{5}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. These equations have real solutions and complex, but not real, solutions. $$ (x+5)(x-1)=2 $$

Use the discriminant to determine the number and types of solutions of each equation. $$ x^{2}-5=0 $$

Factor. $$ z^{4}-13 z^{2}+36 $$

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ \frac{1}{6} x^{2}+x+\frac{1}{3}=0 $$

Solve. Bailey Wilson's rectangular dog pen for her Irish setter must have an area of 400 square feet. Also, the length must be 10 feet longer than the width. Find the dimensions of the pen.
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