What Is The Complete Factorization Of The Polynomial Below?$x^3+2x^2+x+2$A. \[$(x+2)(x+i)(x+i)\$\]B. \[$(x-2)(x+i)(x-i)\$\]C. \[$(x-2)(x+i)(x+i)\$\]D. \[$(x+2)(x+i)(x-i)\$\]

Introduction

In mathematics, polynomial factorization is a crucial concept that helps us simplify complex expressions and solve equations. When we factorize a polynomial, we express it as a product of simpler polynomials, known as factors. In this article, we will explore the complete factorization of the given polynomial $x^3+2x^2+x+2$ and determine the correct answer among the options provided.

Understanding the Polynomial

The given polynomial is a cubic polynomial, which means it has a degree of 3. It can be written in the form $ax^3+bx^2+cx+d$, where $a=1$, $b=2$, $c=1$, and $d=2$. To factorize this polynomial, we need to find the values of $x$ that make the polynomial equal to zero.

Factoring the Polynomial

To factorize the polynomial, we can start by looking for common factors. In this case, there are no common factors, so we need to use other methods to factorize the polynomial. One method is to use the Rational Root Theorem, which states that if a rational number $p/q$ is a root of the polynomial, then $p$ must be a factor of the constant term $d$, and $q$ must be a factor of the leading coefficient $a$.

Applying the Rational Root Theorem

Using the Rational Root Theorem, we can determine the possible rational roots of the polynomial. The factors of the constant term $d=2$ are $\pm 1$ and $\pm 2$, and the factors of the leading coefficient $a=1$ are $\pm 1$. Therefore, the possible rational roots of the polynomial are $\pm 1$ and $\pm 2$.

Finding the Roots of the Polynomial

To find the roots of the polynomial, we can use synthetic division or long division. Let's use synthetic division to divide the polynomial by the possible rational roots. We start by dividing the polynomial by $x-2$, which is one of the possible rational roots.

Synthetic Division

Using synthetic division, we get:

1	2	1	2
-2	0	4	2

The result of the synthetic division is a quotient of $x^2+4$ and a remainder of 2. This means that $x-2$ is a factor of the polynomial, and the quotient $x^2+4$ is also a factor.

Factoring the Quotient

The quotient $x^2+4$ can be factored as $(x+i)(x-i)$, where $i$ is the imaginary unit. This is because $x^2+4$ is a difference of squares, and we can factor it as $(x+i)(x-i)$.

Complete Factorization

Now that we have factored the polynomial, we can write the complete factorization as:

$$x^3+2x^2+x+2 = (x-2)(x+i)(x-i)$$

Conclusion

In conclusion, the complete factorization of the polynomial $x^3+2x^2+x+2$ is $(x-2)(x+i)(x-i)$. This is the correct answer among the options provided.

Final Answer

The final answer is:

• Option B: [$(x-2)(x+i)(x-i)$]

This is the correct answer, and it is the complete factorization of the given polynomial.

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Introduction

In our previous article, we explored the complete factorization of the polynomial $x^3+2x^2+x+2$. We determined that the correct factorization is $(x-2)(x+i)(x-i)$. In this article, we will answer some frequently asked questions related to the factorization of this polynomial.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem is a theorem in algebra that states that if a rational number $p/q$ is a root of a polynomial, then $p$ must be a factor of the constant term $d$, and $q$ must be a factor of the leading coefficient $a$.

Q: How do I apply the Rational Root Theorem to factorize a polynomial?

A: To apply the Rational Root Theorem, you need to determine the factors of the constant term $d$ and the leading coefficient $a$. Then, you can use synthetic division or long division to divide the polynomial by the possible rational roots.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a linear factor. It is a shortcut for long division and can be used to find the roots of a polynomial.

Q: How do I use synthetic division to factorize a polynomial?

A: To use synthetic division, you need to divide the polynomial by the possible rational roots. You can use a table or a diagram to help you with the calculations.

Q: What is the difference between $(x+i)$ and $(x-i)$?

A: $(x+i)$ and $(x-i)$ are conjugate pairs of complex numbers. They are used to factorize polynomials with complex roots.

Q: Why do we need to use conjugate pairs to factorize polynomials?

A: We need to use conjugate pairs to factorize polynomials because complex roots always come in conjugate pairs. This means that if a polynomial has a complex root, it will also have the conjugate of that root.

Q: Can I use the Rational Root Theorem to factorize polynomials with complex roots?

A: Yes, you can use the Rational Root Theorem to factorize polynomials with complex roots. However, you need to be careful when using the theorem, as it only works for rational roots.

Q: How do I determine the correct factorization of a polynomial?

A: To determine the correct factorization of a polynomial, you need to use a combination of the Rational Root Theorem, synthetic division, and factoring techniques. You also need to check your work to make sure that the factorization is correct.

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

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

• Not using the Rational Root Theorem to determine the possible rational roots
• Not using synthetic division to divide the polynomial by the possible rational roots
• Not checking the work to make sure that the factorization is correct
• Not using conjugate pairs to factorize polynomials with complex roots

Conclusion

In conclusion, factorizing polynomials can be a challenging task, but with the right techniques and tools, you can determine the correct factorization. Remember to use the Rational Root Theorem, synthetic division, and factoring techniques to factorize polynomials, and always check your work to make sure that the factorization is correct.

Final Tips

• Practice, practice, practice: The more you practice factorizing polynomials, the more comfortable you will become with the techniques and tools.
• Use technology: There are many online tools and software programs that can help you factorize polynomials.
• Read and understand the problem: Before you start factorizing a polynomial, make sure that you understand the problem and what is being asked.
• Check your work: Always check your work to make sure that the factorization is correct.

By following these tips and using the techniques and tools outlined in this article, you can become proficient in factorizing polynomials and solving equations.