What Is The Completely Factored Form Of $25 X^4 - 16 Y^2$?A. $\left(5 X^4 + 4 Y\right)(5 X - 4 Y$\]B. $\left(5 X^3 + 4 Y\right)\left(5 X^2 - 4 Y\right$\]C. $\left(5 X^2 + 4 Y\right)\left(5 X^2 - 4 Y\right$\]D. $25

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What is the Completely Factored Form of $25 x^4 - 16 y^2$?

The completely factored form of a polynomial expression is a product of its irreducible factors. In other words, it is the expression written as a product of prime factors, where each factor is raised to the power of its multiplicity. Factoring a polynomial expression can be a challenging task, especially when dealing with higher-degree polynomials. However, with the right techniques and strategies, we can factor even the most complex expressions.

Understanding the Given Expression

The given expression is $25 x^4 - 16 y^2$. This is a difference of squares, where the first term is a perfect square and the second term is also a perfect square. The first term can be written as $(5x2)2$, and the second term can be written as $(4y)^2$. This allows us to rewrite the expression as $(5x2)2 - (4y)^2$.

Factoring the Difference of Squares

The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. We can apply this formula to our expression by letting $a = 5x^2$ and $b = 4y$. This gives us $(5x^2 + 4y)(5x^2 - 4y)$.

Checking the Answer Choices

Now that we have factored the expression, we can check the answer choices to see which one matches our result. Let's examine each choice:

A. $\left(5 x^4 + 4 y\right)(5 x - 4 y\right)$

This choice is incorrect because the first term in the first factor is $5x^4$, which is not equal to $(5x2)2$.

B. $\left(5 x^3 + 4 y\right)\left(5 x^2 - 4 y\right)$

This choice is incorrect because the first term in the first factor is $5x^3$, which is not equal to $(5x2)2$.

C. $\left(5 x^2 + 4 y\right)\left(5 x^2 - 4 y\right)$

This choice is correct because it matches our result from factoring the difference of squares.

D. $25 x^4 - 16 y^2$

This choice is incorrect because it is the original expression, not the factored form.

Conclusion

In conclusion, the completely factored form of $25 x^4 - 16 y^2$ is $\left(5 x^2 + 4 y\right)\left(5 x^2 - 4 y\right)$. This result was obtained by recognizing the difference of squares and applying the corresponding formula. We also checked the answer choices to ensure that our result was correct.

Key Takeaways

  • The completely factored form of a polynomial expression is a product of its irreducible factors.
  • The difference of squares formula is a useful tool for factoring expressions of the form $a^2 - b^2$.
  • When factoring a polynomial expression, it is essential to recognize the underlying structure and apply the corresponding formulas.

Additional Examples

Here are a few additional examples of factoring polynomial expressions:

  • 9x2−16y2=(3x+4y)(3x−4y)9x^2 - 16y^2 = (3x + 4y)(3x - 4y)

  • x2−4y2=(x+2y)(x−2y)x^2 - 4y^2 = (x + 2y)(x - 2y)

  • 25x2−9y2=(5x+3y)(5x−3y)25x^2 - 9y^2 = (5x + 3y)(5x - 3y)

These examples demonstrate the importance of recognizing the underlying structure of a polynomial expression and applying the corresponding formulas to factor it.

Real-World Applications

Factoring polynomial expressions has numerous real-world applications in various fields, including:

  • Algebra: Factoring polynomial expressions is a fundamental concept in algebra, and it is used to solve equations and inequalities.
  • Calculus: Factoring polynomial expressions is used to find the derivatives and integrals of functions.
  • Physics: Factoring polynomial expressions is used to solve problems involving motion, energy, and momentum.
  • Engineering: Factoring polynomial expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.

In conclusion, factoring polynomial expressions is a crucial skill that has numerous real-world applications. By recognizing the underlying structure of a polynomial expression and applying the corresponding formulas, we can factor even the most complex expressions.
Q&A: Factoring Polynomial Expressions

In this article, we will answer some common questions related to factoring polynomial expressions. Whether you are a student, a teacher, or a professional, this article will provide you with the information you need to understand and apply factoring techniques.

Q: What is factoring a polynomial expression?

A: Factoring a polynomial expression is the process of expressing it as a product of its irreducible factors. In other words, it is the process of breaking down a polynomial expression into simpler expressions that cannot be factored further.

Q: Why is factoring important?

A: Factoring is important because it allows us to simplify polynomial expressions, which can make them easier to work with. It also helps us to identify the roots of a polynomial equation, which is essential in many areas of mathematics and science.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Greatest Common Factor (GCF) factoring: This involves factoring out the greatest common factor of two or more terms.
  • Difference of Squares factoring: This involves factoring expressions of the form $a^2 - b^2$.
  • Sum and Difference factoring: This involves factoring expressions of the form $a^2 + 2ab + b^2$ and $a^2 - 2ab + b^2$.
  • Cubic factoring: This involves factoring expressions of the form $ax^3 + bx^2 + cx + d$.

Q: How do I factor a polynomial expression?

A: Factoring a polynomial expression involves several steps:

  1. Identify the type of factoring: Determine which type of factoring is required, such as GCF factoring, Difference of Squares factoring, or Sum and Difference factoring.
  2. Look for common factors: Look for common factors among the terms, such as a greatest common factor or a common binomial factor.
  3. Apply the factoring formula: Apply the corresponding factoring formula to the expression.
  4. Simplify the expression: Simplify the expression by combining like terms.

Q: What are some common mistakes to avoid when factoring?

A: Some common mistakes to avoid when factoring include:

  • Not identifying the type of factoring: Failing to identify the type of factoring required can lead to incorrect results.
  • Not looking for common factors: Failing to look for common factors can lead to missing out on simplifications.
  • Not applying the factoring formula correctly: Applying the factoring formula incorrectly can lead to incorrect results.
  • Not simplifying the expression: Failing to simplify the expression can lead to unnecessary complexity.

Q: How do I check my factoring?

A: To check your factoring, you can:

  • Multiply the factors: Multiply the factors to ensure that they produce the original expression.
  • Check for common factors: Check for common factors among the terms to ensure that they have been factored correctly.
  • Simplify the expression: Simplify the expression to ensure that it is in its simplest form.

Q: What are some real-world applications of factoring?

A: Factoring has numerous real-world applications, including:

  • Algebra: Factoring is used to solve equations and inequalities.
  • Calculus: Factoring is used to find the derivatives and integrals of functions.
  • Physics: Factoring is used to solve problems involving motion, energy, and momentum.
  • Engineering: Factoring is used to design and optimize systems, such as electrical circuits and mechanical systems.

In conclusion, factoring polynomial expressions is a crucial skill that has numerous real-world applications. By understanding the different types of factoring and how to apply them, you can simplify polynomial expressions and solve problems in various areas of mathematics and science.