Factor The Following Polynomial Completely:$-x^2 Y^2 + X^4 + 9y^2 - 9x^2$A. \[$(x-3)(x-3)(x+y)(x-y)\$\]B. \[$(x+3)(x+3)(x+y)(x-y)\$\]C. \[$(x+3)(x-3)(x+y)(x-y)\$\]

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Introduction

In algebra, factoring polynomials is an essential skill that helps us simplify complex expressions and solve equations. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Factoring a polynomial involves expressing it as a product of simpler polynomials, known as factors. In this article, we will focus on factoring the given polynomial completely.

The Given Polynomial

The given polynomial is:

−x2y2+x4+9y2−9x2-x^2 y^2 + x^4 + 9y^2 - 9x^2

Our goal is to factor this polynomial completely.

Step 1: Grouping Terms

To factor the polynomial, we can start by grouping the terms that have common factors. We can group the first two terms and the last two terms:

(−x2y2+x4)+(9y2−9x2)(-x^2 y^2 + x^4) + (9y^2 - 9x^2)

Step 2: Factoring Out Common Factors

Now, we can factor out common factors from each group:

x2(x2−y2)+9(y2−x2)x^2(x^2 - y^2) + 9(y^2 - x^2)

Step 3: Factoring the Difference of Squares

We can recognize that the expressions inside the parentheses are difference of squares. We can factor them as follows:

x2(x−y)(x+y)+9(x−y)(x+y)x^2(x - y)(x + y) + 9(x - y)(x + y)

Step 4: Factoring Out the Common Binomial

Now, we can factor out the common binomial (x−y)(x+y)(x - y)(x + y) from both terms:

(x−y)(x+y)(x2+9)(x - y)(x + y)(x^2 + 9)

Step 5: Factoring the Trinomial

The expression x2+9x^2 + 9 is a trinomial that can be factored as a difference of squares:

(x−3)(x+3)(x−y)(x+y)(x - 3)(x + 3)(x - y)(x + y)

Conclusion

After following the steps, we have factored the given polynomial completely. The final answer is:

(x−3)(x+3)(x−y)(x+y)(x - 3)(x + 3)(x - y)(x + y)

This is option A.

Comparison with Other Options

Let's compare our answer with the other options:

  • Option B: (x+3)(x+3)(x+y)(x−y)(x + 3)(x + 3)(x + y)(x - y)
  • Option C: (x+3)(x−3)(x+y)(x−y)(x + 3)(x - 3)(x + y)(x - y)

Our answer is different from both options B and C. This is because the correct factorization of the polynomial requires the correct signs in the factors.

Importance of Factoring Polynomials

Factoring polynomials is an essential skill in algebra that helps us simplify complex expressions and solve equations. It is used in various fields, including physics, engineering, and computer science. By factoring polynomials, we can:

  • Simplify complex expressions
  • Solve equations
  • Find the roots of polynomials
  • Analyze the behavior of functions

Real-World Applications

Factoring polynomials has many real-world applications, including:

  • Physics: Factoring polynomials is used to describe the motion of objects under the influence of forces.
  • Engineering: Factoring polynomials is used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
  • Computer Science: Factoring polynomials is used in cryptography, coding theory, and other areas of computer science.

Conclusion

Introduction

In our previous article, we discussed how to factor the polynomial −x2y2+x4+9y2−9x2-x^2 y^2 + x^4 + 9y^2 - 9x^2 completely. In this article, we will answer some frequently asked questions about factoring polynomials.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, known as factors. This helps us simplify complex expressions and solve equations.

Q: What are the steps to factor a polynomial?

A: The steps to factor a polynomial are:

  1. Group the terms that have common factors.
  2. Factor out common factors from each group.
  3. Recognize and factor the difference of squares.
  4. Factor out the common binomial.
  5. Factor the trinomial.

Q: What is a difference of squares?

A: A difference of squares is an expression of the form a2−b2a^2 - b^2, which can be factored as (a−b)(a+b)(a - b)(a + b).

Q: How do I recognize a difference of squares?

A: To recognize a difference of squares, look for expressions of the form a2−b2a^2 - b^2. You can also use the formula (a−b)(a+b)(a - b)(a + b) to factor the expression.

Q: What is a trinomial?

A: A trinomial is an expression with three terms, such as x2+9x^2 + 9.

Q: How do I factor a trinomial?

A: To factor a trinomial, look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term. Then, factor the trinomial as the product of two binomials.

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

A: Some common mistakes to avoid when factoring polynomials include:

  • Not grouping the terms correctly
  • Not factoring out common factors
  • Not recognizing the difference of squares
  • Not factoring the trinomial correctly

Q: How do I check if my factored form is correct?

A: To check if your factored form is correct, multiply the factors together and simplify the expression. If the result is the original polynomial, then your factored form is correct.

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

A: Factoring polynomials has many real-world applications, including:

  • Physics: Factoring polynomials is used to describe the motion of objects under the influence of forces.
  • Engineering: Factoring polynomials is used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
  • Computer Science: Factoring polynomials is used in cryptography, coding theory, and other areas of computer science.

Conclusion

In conclusion, factoring polynomials is an essential skill in algebra that has many real-world applications. By understanding the steps to factor a polynomial, recognizing the difference of squares, and factoring the trinomial, you can simplify complex expressions and solve equations. Remember to check your factored form by multiplying the factors together and simplifying the expression.