Now, Suppose One Of The Roots Of The Polynomial Function Is Complex. The Roots Of The Function Are $2+i$ And 5. Write The Equation For This Polynomial Function.Which Of The Following Must Also Be A Root Of The Function?A. $-3$B.

Introduction

In mathematics, polynomial functions are a fundamental concept in algebra, and understanding their roots is crucial for solving various mathematical problems. When dealing with polynomial functions, we often encounter real roots, which are values that satisfy the equation. However, in some cases, we may come across complex roots, which are values that involve imaginary numbers. In this article, we will explore the concept of complex roots of polynomial functions and how to determine the equation of a polynomial function given its roots.

Complex Roots and Conjugate Roots

A complex root of a polynomial function is a value that involves an imaginary number, denoted by the letter i. For example, the complex root $2+i$ means that the value of the function is equal to $2+i$ when the variable is equal to a certain value. When a polynomial function has a complex root, it also has a conjugate root, which is obtained by changing the sign of the imaginary part. In the case of the complex root $2+i$, the conjugate root is $2-i$.

The Given Roots

We are given that the roots of the polynomial function are $2+i$ and 5. Since $2+i$ is a complex root, its conjugate root $2-i$ must also be a root of the function. Therefore, we have two roots: $2+i$ and $2-i$.

The Equation of the Polynomial Function

To determine the equation of the polynomial function, we need to find the corresponding factors. Since the roots are $2+i$ and $2-i$, the corresponding factors are $(x-(2+i))$ and $(x-(2-i))$. Multiplying these factors together, we get:

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

Expanding the right-hand side, we get:

$$(x-2-i)(x-2+i) = x^2 - 2x - ix - 2x + 4 + 2i + ix - 2i + i^2$$

Simplifying the expression, we get:

$$x^2 - 4x + 4 + i^2$$

Since $i^2 = -1$, we can substitute this value into the expression:

$$x^2 - 4x + 4 - 1$$

Simplifying further, we get:

$$x^2 - 4x + 3$$

Therefore, the equation of the polynomial function is $x^2 - 4x + 3$.

Must Also Be a Root of the Function

Now, let's consider the options given in the problem. We need to determine which of the following must also be a root of the function.

A. $-3$

To determine if $-3$ is a root of the function, we can substitute this value into the equation:

$$(-3)^2 - 4(-3) + 3 = 9 + 12 + 3 = 24$$

Since $24$ is not equal to $0$, $-3$ is not a root of the function.

B. $3$

To determine if $3$ is a root of the function, we can substitute this value into the equation:

$$(3)^2 - 4(3) + 3 = 9 - 12 + 3 = 0$$

Since $0$ is equal to $0$, $3$ is a root of the function.

Conclusion

In conclusion, we have determined the equation of the polynomial function given its roots. We have also identified which of the given options must also be a root of the function. The equation of the polynomial function is $x^2 - 4x + 3$, and the value that must also be a root of the function is $3$.

Introduction

In our previous article, we explored the concept of complex roots of polynomial functions and how to determine the equation of a polynomial function given its roots. In this article, we will answer some frequently asked questions related to complex roots of polynomial functions.

Q: What is a complex root of a polynomial function?

A: A complex root of a polynomial function is a value that involves an imaginary number, denoted by the letter i. For example, the complex root $2+i$ means that the value of the function is equal to $2+i$ when the variable is equal to a certain value.

Q: What is the conjugate root of a complex root?

A: The conjugate root of a complex root is obtained by changing the sign of the imaginary part. For example, the conjugate root of $2+i$ is $2-i$.

Q: How do I determine the equation of a polynomial function given its roots?

A: To determine the equation of a polynomial function given its roots, you need to find the corresponding factors. Since the roots are $2+i$ and $2-i$, the corresponding factors are $(x-(2+i))$ and $(x-(2-i))$. Multiplying these factors together, you get:

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

Expanding the right-hand side, you get:

$$(x-2-i)(x-2+i) = x^2 - 2x - ix - 2x + 4 + 2i + ix - 2i + i^2$$

Simplifying the expression, you get:

$$x^2 - 4x + 4 + i^2$$

Since $i^2 = -1$, you can substitute this value into the expression:

$$x^2 - 4x + 4 - 1$$

Simplifying further, you get:

$$x^2 - 4x + 3$$

Therefore, the equation of the polynomial function is $x^2 - 4x + 3$.

Q: What is the relationship between complex roots and conjugate roots?

A: When a polynomial function has a complex root, it also has a conjugate root, which is obtained by changing the sign of the imaginary part. This is because complex roots always come in conjugate pairs.

Q: How do I determine if a value is a root of a polynomial function?

A: To determine if a value is a root of a polynomial function, you can substitute the value into the equation and check if the result is equal to 0. If the result is equal to 0, then the value is a root of the function.

Q: What is the significance of complex roots in polynomial functions?

A: Complex roots are significant in polynomial functions because they can be used to determine the equation of the function. Additionally, complex roots can be used to solve systems of equations and to find the solutions to quadratic equations.

Q: Can complex roots be real numbers?

A: No, complex roots cannot be real numbers. Complex roots involve imaginary numbers, which are denoted by the letter i.

Q: Can a polynomial function have only real roots?

A: Yes, a polynomial function can have only real roots. However, if a polynomial function has a complex root, it also has a conjugate root.

Q: Can a polynomial function have only complex roots?

A: No, a polynomial function cannot have only complex roots. A polynomial function must have at least one real root.

Q: How do I find the complex roots of a polynomial function?

A: To find the complex roots of a polynomial function, you can use the quadratic formula or the factoring method. The quadratic formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The factoring method involves finding the factors of the polynomial function and then solving for the roots.

Q: Can complex roots be used to solve systems of equations?

A: Yes, complex roots can be used to solve systems of equations. Complex roots can be used to find the solutions to quadratic equations and to solve systems of linear equations.

Q: Can complex roots be used to find the equation of a polynomial function?

A: Yes, complex roots can be used to find the equation of a polynomial function. By finding the corresponding factors, you can multiply them together to get the equation of the polynomial function.

Q: What is the relationship between complex roots and the degree of a polynomial function?

A: The degree of a polynomial function is the highest power of the variable in the polynomial function. Complex roots can be used to determine the degree of a polynomial function.

Q: Can complex roots be used to find the solutions to quadratic equations?

A: Yes, complex roots can be used to find the solutions to quadratic equations. By using the quadratic formula or the factoring method, you can find the complex roots of a quadratic equation.

Q: Can complex roots be used to solve systems of linear equations?

A: Yes, complex roots can be used to solve systems of linear equations. By using the factoring method or the quadratic formula, you can find the complex roots of a system of linear equations.

Conclusion

In conclusion, complex roots of polynomial functions are an important concept in mathematics. By understanding the concept of complex roots, you can determine the equation of a polynomial function given its roots and solve systems of equations. Additionally, complex roots can be used to find the solutions to quadratic equations and to solve systems of linear equations.