Question

\(\left(x^{2}-4\right)\left(x^{2}+4\right)=\) ____.

Short answer

(x-2)(x+2)(x^2+4)

Step-by-step solution

- Recognize the Form

Notice that the expression \((x^{2}-4)\text{ and }(x^{2}+4)\) involves terms that can be related to the difference of squares. Recall that \((a^{2}-b^{2})=(a-b)(a+b)\).

- Apply the Difference of Squares

Recognize that \(4=2^{2}\), thus \(x^{2}-4=x^{2}-2^{2}=(x-2)(x+2)\). The term \(x^{2}+4\) cannot be factored further using real numbers.

- Combine the Factors

Substitute back into the original expression: \((x^{2}-4)(x^{2}+4)=(x-2)(x+2)(x^{2}+4)\).

- Final Simplification

Since \(x^2 + 4\) cannot be simplified further and does not factor into real numbers, the final expanded form of the expression is \((x-2)(x+2)(x^2+4)\). No further simplification is required.

Key concepts

difference of squares

The 'difference of squares' is one of the key concepts in algebra. It ensures we understand how to break down expressions like \(a^2 - b^2\) into simpler terms. The formula for the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\).
This technique applies when you see two squares separated by a subtraction sign. For instance, in the original exercise \((x^2 - 4)(x^2 + 4))\), we can identify \(x^2 - 4\) as a difference of squares because 4 is the same as \(2^2\).
As a result, \((x^2 - 4)\) becomes \((x - 2)(x + 2)\). Recognizing this helps in simplifying and solving polynomial expressions.
So remember, whenever you encounter an expression with two square terms separated by a minus sign, think about the difference of squares.

factoring techniques

Factoring techniques are essential tools in algebra that help break down complex expressions into simpler pieces. They include various methods, such as:

• Difference of squares
• Factoring out common factors
• Factoring trinomials
• Factoring by grouping

In the exercise, we used the difference of squares to factor \((x^2 - 4)\). But remember, \(x^2 + 4\) could not be factored further using real numbers. Thus, understanding when to use which method is crucial.
Always start by looking for common factors. Then, apply more specific techniques like difference of squares, if applicable. The goal is to simplify the expression as much as possible. Practice makes perfect, so keep working on different polynomial problems to master these techniques!

simplification of expressions

The simplification of expressions involves breaking down a complex expression into its simplest form. This not only makes it easier to work with but also helps in solving equations.
In the given exercise, after using the difference of squares, our expression \((x^2 - 4)(x^2 + 4))\) was simplified to \((x - 2)(x + 2)(x^2 + 4)\).
Simplification is crucial because it converts an intricate expression into an understandable one.
Here are the key steps for simplification:

• Identify and factor out any common factors first.
• Look for special products like the difference of squares, perfect square trinomials, or sum/difference of cubes.
• Combine like terms where possible.

By following these steps, we ensure that our expressions are as simple as they can be. This makes solving equations or analyzing their behavior much more straightforward.
Keep practicing different problems to get better at identifying which simplification technique to use!