Factoring Complex and Real Factors

Oct 13, 2025 by windailys.com 128 views

Introduction

In algebra, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When it comes to factoring polynomials, we often encounter various types of factors, including real and complex numbers. In this article, we will delve into the world of polynomial factoring, focusing on a polynomial with a leading coefficient of 1 and specific factors with multiplicity 1.

Understanding the Problem

The given polynomial has a leading coefficient of 1, which means the polynomial is in the form of $ax^n + bx^{n-1} + \ldots + c$ , where $a = 1$ . Additionally, the polynomial has two factors with multiplicity 1: $x - (2+i)$ and $x - \sqrt{2}$ . Our goal is to find the factored form of this polynomial.

Factoring Complex and Real Factors

To factor the given polynomial, we need to consider both the complex and real factors. The complex factor $x - (2+i)$ can be factored using the difference of squares formula, which states that $a^2 - b^2 = (a - b)(a + b)$ . In this case, we have:

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

To factor this expression, we can multiply it by its conjugate, which is $(x - 2) + i$ . This will eliminate the imaginary part and give us a real factor.

Factoring the Complex Factor

Using the conjugate multiplication method, we can factor the complex factor as follows:

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

Expanding the right-hand side, we get:

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

Since $i^2 = -1$ , we can simplify the expression to:

$(x - 2)^2 + 1$

Factoring the Real Factor

Now that we have factored the complex factor, we can focus on the real factor $x - \sqrt{2}$ . This factor can be factored using the difference of squares formula as well:

$x - \sqrt{2} = (x - \sqrt{2})(x + \sqrt{2})$

Combining the Factors

Now that we have factored both the complex and real factors, we can combine them to find the factored form of the polynomial. Multiplying the factored complex factor by the factored real factor, we get:

$(x - (2+i))(x - (2-i))(x - \sqrt{2})(x + \sqrt{2})$

Conclusion

In this article, we have factored a polynomial with a leading coefficient of 1 and specific factors with multiplicity 1. We have used the difference of squares formula to factor both the complex and real factors, and then combined them to find the factored form of the polynomial. The final answer is:

A. ${x-(2+i)][x-(2-i)](x-\sqrt{2})(x+\sqrt{2}$ }$

This is the correct factored form of the polynomial.

Final Answer

The final answer is A. ${x-(2+i)][x-(2-i)](x-\sqrt{2})(x+\sqrt{2}$ }$.

Introduction

In our previous article, we explored the concept of factoring polynomials with complex and real factors. We delved into the world of polynomial factoring, focusing on a polynomial with a leading coefficient of 1 and specific factors with multiplicity 1. In this article, we will continue the discussion by answering some frequently asked questions related to polynomial factoring.

Q&A

Q: What is the difference between a complex factor and a real factor?

A: A complex factor is a factor that contains an imaginary number, such as $i$ or $-i$ . On the other hand, a real factor is a factor that contains only real numbers.

Q: How do I factor a complex factor using the conjugate multiplication method?

A: To factor a complex factor using the conjugate multiplication method, you need to multiply the factor by its conjugate. The conjugate of a complex number $a + bi$ is $a - bi$ . For example, if you have the complex factor $x - (2+i)$ , you can multiply it by its conjugate $(x - 2) + i$ to eliminate the imaginary part.

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that 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 factor a real factor using the difference of squares formula?

A: To factor a real factor using the difference of squares formula, you need to identify the expression in the form $a^2 - b^2$ . For example, if you have the real factor $x - \sqrt{2}$ , you can factor it using the difference of squares formula as $(x - \sqrt{2})(x + \sqrt{2})$ .

Q: What is the factored form of the polynomial $x^2 - 4x + 5$ ?

A: To factor the polynomial $x^2 - 4x + 5$ , you need to find two numbers whose product is $5$ and whose sum is $-4$ . These numbers are $-1$ and $-5$ . Therefore, the factored form of the polynomial is $(x - 1)(x - 5)$ .

Q: How do I determine the multiplicity of a factor?

A: The multiplicity of a factor is the number of times the factor appears in the polynomial. For example, if you have the polynomial $x^3 - 2x^2 + 3x - 1$ , the factor $x - 1$ appears three times, so its multiplicity is $3$ .

Q: What is the relationship between the degree of a polynomial and its factors?

A: The degree of a polynomial is the highest power of the variable in the polynomial. The factors of a polynomial are the expressions that, when multiplied together, give the polynomial. The degree of a polynomial is equal to the sum of the multiplicities of its factors.

Conclusion

In this article, we have answered some frequently asked questions related to polynomial factoring. We have discussed the difference between complex and real factors, the conjugate multiplication method, the difference of squares formula, and the relationship between the degree of a polynomial and its factors. We hope that this article has provided you with a better understanding of polynomial factoring and its applications.

Final Answer

The final answer is that the factored form of the polynomial is A. ${x-(2+i)][x-(2-i)](x-\sqrt{2})(x+\sqrt{2}$ }$.