Factor The Expression Completely.$\[ 16 - 40x^4 \\]

Introduction

Factoring an expression is a fundamental concept in algebra that involves breaking down a complex expression into simpler components. In this article, we will focus on factoring the expression $16 - 40x^4$ completely. Factoring expressions is a crucial skill in mathematics, as it allows us to simplify complex equations, identify patterns, and solve problems more efficiently.

Understanding the Expression

Before we begin factoring the expression, let's take a closer look at its components. The expression $16 - 40x^4$ consists of two terms: a constant term $16$ and a variable term $-40x^4$. The variable term is a polynomial of degree $4$, which means it has four terms with different powers of $x$.

Factoring Out the Greatest Common Factor (GCF)

One of the first steps in factoring an expression is to identify the greatest common factor (GCF) of the terms. In this case, the GCF of $16$ and $-40x^4$ is $-8$. We can factor out the GCF by dividing each term by $-8$.

16 - 40x^4 = -8(2 + 5x^4)

Factoring the Quadratic Expression

Now that we have factored out the GCF, we are left with a quadratic expression $2 + 5x^4$. This expression can be factored further by recognizing that it is a difference of squares.

2 + 5x^4 = (1 + x^2)^2 - (2x^2)^2

Applying the Difference of Squares Formula

The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. We can apply this formula to the quadratic expression by letting $a = 1 + x^2$ and $b = 2x^2$.

(1 + x^2)^2 - (2x^2)^2 = (1 + x^2 + 2x^2)(1 + x^2 - 2x^2)

Simplifying the Expression

Now that we have applied the difference of squares formula, we can simplify the expression by combining like terms.

(1 + x^2 + 2x^2)(1 + x^2 - 2x^2) = (3x^2 + 1)(1 - x^2)

Factoring the Final Expression

We are now left with a final expression $(3x^2 + 1)(1 - x^2)$. This expression cannot be factored further, as it is already in its simplest form.

Conclusion

In this article, we have factored the expression $16 - 40x^4$ completely. We began by factoring out the greatest common factor (GCF) of the terms, and then applied the difference of squares formula to factor the quadratic expression. Finally, we simplified the expression by combining like terms. Factoring expressions is a crucial skill in mathematics, and this article has provided a step-by-step guide on how to factor the expression completely.

Common Mistakes to Avoid

When factoring expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not factoring out the GCF: Failing to factor out the greatest common factor (GCF) can lead to incorrect factorization.
• Not recognizing the difference of squares: Failing to recognize the difference of squares formula can lead to incorrect factorization.
• Not simplifying the expression: Failing to simplify the expression by combining like terms can lead to incorrect factorization.

Real-World Applications

Factoring expressions has many real-world applications. Here are a few examples:

• Simplifying complex equations: Factoring expressions can help simplify complex equations, making them easier to solve.
• Identifying patterns: Factoring expressions can help identify patterns in mathematics, which can lead to new discoveries.
• Solving problems more efficiently: Factoring expressions can help solve problems more efficiently, as it allows us to break down complex expressions into simpler components.

Final Thoughts

Introduction

Factoring expressions is a fundamental concept in algebra that involves breaking down a complex expression into simpler components. In our previous article, we provided a step-by-step guide on how to factor the expression $16 - 40x^4$ completely. In this article, we will answer some frequently asked questions (FAQs) about factoring expressions.

Q: What is factoring an expression?

A: Factoring an expression involves breaking down a complex expression into simpler components, such as polynomials or other algebraic expressions.

Q: Why is factoring an expression important?

A: Factoring an expression is important because it allows us to simplify complex equations, identify patterns, and solve problems more efficiently.

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