What Is The Completely Factored Form Of $x^4 + 8x^2 - 9$?A. $(x+1)(x-1)(x+3)(x+3)$ B. $(x+1)(x-1)\left(x^2+9\right)$ C. \$\left(x^2-1\right)(x+3)(x-3)$[/tex\] D. $(x+1)(x+1)(x+3)(x+3)$
Introduction
In mathematics, factoring polynomials is a crucial concept that helps us simplify complex expressions and solve equations. When it comes to factoring a polynomial of degree 4, we need to find two quadratic expressions whose product is the given polynomial. In this article, we will explore the completely factored form of the polynomial $x^4 + 8x^2 - 9$ and examine the different options provided.
Understanding the Polynomial
The given polynomial is a quartic polynomial, which means it has a degree of 4. It can be written as $x^4 + 8x^2 - 9$. To factor this polynomial, we need to find two quadratic expressions whose product is equal to the given polynomial.
Factoring the Polynomial
To factor the polynomial, we can start by looking for two numbers whose product is -9 and whose sum is 8. These numbers are 9 and -1, since 9 × (-1) = -9 and 9 + (-1) = 8. However, we need to find two quadratic expressions whose product is the given polynomial.
Using the Difference of Squares Formula
We can use the difference of squares formula to factor the polynomial. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. We can rewrite the polynomial as $(x2)2 - 9$, which is a difference of squares.
Applying the Difference of Squares Formula
Applying the difference of squares formula, we get $(x2)2 - 9 = (x^2 + 3)(x^2 - 3)$. However, this is not the completely factored form of the polynomial.
Factoring the Quadratic Expressions
We can factor the quadratic expressions further by using the difference of squares formula again. We get $(x^2 + 3) = (x + \sqrt{3})(x - \sqrt{3})$ and $(x^2 - 3) = (x + \sqrt{3})(x - \sqrt{3})$.
Finding the Completely Factored Form
However, we notice that the quadratic expressions $(x^2 + 3)$ and $(x^2 - 3)$ have a common factor of $(x + \sqrt{3})$ and $(x - \sqrt{3})$. We can factor out this common factor to get the completely factored form of the polynomial.
The Completely Factored Form
The completely factored form of the polynomial $x^4 + 8x^2 - 9$ is $(x^2 + 3)(x^2 - 3)$. However, we can factor the quadratic expressions further by using the difference of squares formula again.
Factoring the Quadratic Expressions Further
We get $(x^2 + 3) = (x + \sqrt{3})(x - \sqrt{3})$ and $(x^2 - 3) = (x + \sqrt{3})(x - \sqrt{3})$. However, we notice that the quadratic expressions $(x^2 + 3)$ and $(x^2 - 3)$ have a common factor of $(x + \sqrt{3})$ and $(x - \sqrt{3})$.
The Completely Factored Form of the Polynomial
The completely factored form of the polynomial $x^4 + 8x^2 - 9$ is $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$.
Conclusion
In this article, we have explored the completely factored form of the polynomial $x^4 + 8x^2 - 9$. We have used the difference of squares formula to factor the polynomial and have found the completely factored form of the polynomial to be $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$.
Answer
The completely factored form of the polynomial $x^4 + 8x^2 - 9$ is $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$.
Comparison with Options
We can compare our answer with the options provided. Option A is $(x+1)(x-1)(x+3)(x+3)$, which is not the completely factored form of the polynomial. Option B is $(x+1)(x-1)\left(x^2+9\right)$, which is not the completely factored form of the polynomial. Option C is $\left(x^2-1\right)(x+3)(x-3)$, which is not the completely factored form of the polynomial. Option D is $(x+1)(x+1)(x+3)(x+3)$, which is not the completely factored form of the polynomial.
Final Answer
The completely factored form of the polynomial $x^4 + 8x^2 - 9$ is $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$.
Discussion
The completely factored form of the polynomial $x^4 + 8x^2 - 9$ is a crucial concept in mathematics. It helps us simplify complex expressions and solve equations. In this article, we have explored the completely factored form of the polynomial and have found the completely factored form of the polynomial to be $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$.
Conclusion
In conclusion, the completely factored form of the polynomial $x^4 + 8x^2 - 9$ is $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$. This is a crucial concept in mathematics that helps us simplify complex expressions and solve equations.
References
- [1] "Factoring Polynomials" by Math Open Reference
- [2] "Difference of Squares Formula" by Math Is Fun
- [3] "Quadratic Expressions" by Khan Academy
Keywords
- Factoring polynomials
- Difference of squares formula
- Quadratic expressions
- Completely factored form
- Mathematics
Introduction
In our previous article, we explored the completely factored form of the polynomial $x^4 + 8x^2 - 9$. We used the difference of squares formula to factor the polynomial and found the completely factored form of the polynomial to be $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$. In this article, we will answer some frequently asked questions about the completely factored form of the polynomial.
Q: What is the completely factored form of the polynomial $x^4 + 8x^2 - 9$?
A: The completely factored form of the polynomial $x^4 + 8x^2 - 9$ is $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$.
Q: How do I factor the polynomial $x^4 + 8x^2 - 9$?
A: To factor the polynomial $x^4 + 8x^2 - 9$, you can use the difference of squares formula. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. You can rewrite the polynomial as $(x2)2 - 9$, which is a difference of squares.
Q: What is the difference of squares formula?
A: The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to factor expressions of the form $a^2 - b^2$.
Q: How do I use the difference of squares formula to factor the polynomial $x^4 + 8x^2 - 9$?
A: To use the difference of squares formula to factor the polynomial $x^4 + 8x^2 - 9$, you can rewrite the polynomial as $(x2)2 - 9$. Then, you can apply the difference of squares formula to get $(x^2 + 3)(x^2 - 3)$.
Q: What is the completely factored form of the polynomial $(x^2 + 3)(x^2 - 3)$?
A: The completely factored form of the polynomial $(x^2 + 3)(x^2 - 3)$ is $(x + \sqrt{3})(x - \sqrt{3})(x + \sqrt{3})(x - \sqrt{3})$.
Q: Why is the completely factored form of the polynomial $x^4 + 8x^2 - 9$ important?
A: The completely factored form of the polynomial $x^4 + 8x^2 - 9$ is important because it helps us simplify complex expressions and solve equations. By factoring the polynomial, we can make it easier to work with and solve problems involving the polynomial.
Q: Can I use the completely factored form of the polynomial $x^4 + 8x^2 - 9$ to solve equations?
A: Yes, you can use the completely factored form of the polynomial $x^4 + 8x^2 - 9$ to solve equations. By factoring the polynomial, you can make it easier to work with and solve problems involving the polynomial.
Q: What are some common mistakes to avoid when factoring the polynomial $x^4 + 8x^2 - 9$?
A: Some common mistakes to avoid when factoring the polynomial $x^4 + 8x^2 - 9$ include:
- Not using the difference of squares formula
- Not rewriting the polynomial as $(x2)2 - 9$
- Not applying the difference of squares formula correctly
- Not factoring the polynomial completely
Conclusion
In this article, we have answered some frequently asked questions about the completely factored form of the polynomial $x^4 + 8x^2 - 9$. We have discussed how to factor the polynomial using the difference of squares formula and have provided some common mistakes to avoid when factoring the polynomial.
References
- [1] "Factoring Polynomials" by Math Open Reference
- [2] "Difference of Squares Formula" by Math Is Fun
- [3] "Quadratic Expressions" by Khan Academy
Keywords
- Factoring polynomials
- Difference of squares formula
- Quadratic expressions
- Completely factored form
- Mathematics