Problem 49 - Elementary and Intermediate Algebra

Question

Find each special product. $$ \left(s^{2}+4\right)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \( s^4 + 8s^2 + 16 \).

Step 1: Identify the Formula to Use

The expression \( \left(s^2 + 4\right)^2 \) is a perfect square trinomial. We can expand it using the square of a binomial formula: \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = s^2\) and \(b = 4\).

Step 2: Apply the Formula

Using \((a+b)^2 = a^2 + 2ab + b^2\), substitute \(a = s^2\) and \(b = 4\):\[ (s^2 + 4)^2 = (s^2)^2 + 2 \cdot s^2 \cdot 4 + 4^2 \]

Step 3: Calculate Each Term Separately

Now calculate each part of the expanded formula:- \((s^2)^2 = s^4\)- \(2 \cdot s^2 \cdot 4 = 8s^2\)- \(4^2 = 16\)

Step 4: Combine the Terms

Add the resulting terms together:\[ s^4 + 8s^2 + 16 \]

Step 5: Simplify the Expression

In this step, we verify if the terms can be further simplified. Since they are already simplified, our final expression is:\[ s^4 + 8s^2 + 16 \]

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that can be written in the form \((a + b)^2\). It happens when a binomial is squared, leading to a specific pattern of terms. For example, when squaring \(s^2 + 4\), you get a polynomial in which every term is crafted from mixing and matching the squared terms and their products from the original binomial. This structure is important because it simplifies the process of expansion, using the formula \((a + b)^2 = a^2 + 2ab + b^2\) to find our final polynomial form with ease.

Recognizing a perfect square trinomial quickly is valuable in algebra because it reduces the need for lengthy multiplication. When you see something like \(s^2 + 4\) followed by a square, it should hint you at the pattern you need to follow to expand it effortlessly. You can immediately apply the formula without breaking down every multiplication step, saving you time and making calculations straightforward.

Binomial Expansion

Binomial expansion is the process of expanding expressions involving binomials, like \(s^2 + 4\). The beauty of this mathematical concept lies in the formula \((a + b)^2\). With this, you can expand the expression into a polynomial form readily. Rather than multiplying each term individually, binomial expansion allows us to easily identify the resulting pattern.

First, in our example with \(s^2 + 4\), you assign \(a = s^2\) and \(b = 4\). Then plug these into the formula: \((s^2)^2 + 2 (s^2) (4) + 4^2\). The algebraic procedure involves squaring and multiplying these components according to the rule, which results in new terms that are combined to form a polynomial.

• The square of the first term: \(s^4\)
• Twice the product of both terms: \(8s^2\)
• The square of the second term: \(16\)

This method illustrates a basic principle in algebra that reapply in numerous scenarios, helping you tackle problems more effectively.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial to its simplest form. After calculating terms individually, like in the transformations we performed with \(s^4 + 8s^2 + 16\), your goal is to combine like terms and validate that no further simplification is needed.

Polynomials are simplified by ensuring that there are no more like terms to combine, and all arithmetic within the polynomial is as reduced as possible. In the example provided, each term has a distinct variable power or constant value, which means there are no like terms to combine. This reveals that \(s^4 + 8s^2 + 16\) is already in its simplest form.

Understanding this process is crucial for algebra, as it not only helps in solving equations or factoring polynomials but also ensures the results are as concise as possible. It teaches students to check their work, affirming mathematical accuracy and boosting confidence in handling algebraic expressions.