Given That 3 Is A Zero Of The Polynomial Function $f(x)$, Find The Remaining Zeros.$f(x) = X^3 - 7x^2 + 20x - 24$List The Remaining Zeros (other Than 3). $\square$(Simplify Your Answer. Type An Exact Answer, Using Radicals

Introduction

When given a polynomial function and one of its zeros, we can use various methods to find the remaining zeros. In this article, we will focus on finding the remaining zeros of a cubic polynomial function, given that one of its zeros is 3. We will use the given zero to factorize the polynomial and then find the remaining zeros.

Given Polynomial Function

The given polynomial function is $f(x) = x^3 - 7x^2 + 20x - 24$. We are given that 3 is a zero of this polynomial function.

Factoring the Polynomial

To find the remaining zeros, we can start by factoring the polynomial using the given zero. Since 3 is a zero, we know that $(x - 3)$ is a factor of the polynomial. We can use polynomial division or synthetic division to divide the polynomial by $(x - 3)$.

Polynomial Division

Using polynomial division, we can divide the polynomial $f(x) = x^3 - 7x^2 + 20x - 24$ by $(x - 3)$. The result of the division is:

$$\frac{x^3 - 7x^2 + 20x - 24}{x - 3} = x^2 - 4x + 8$$

Quadratic Factor

The result of the division is a quadratic factor $(x^2 - 4x + 8)$. We can use the quadratic formula to find the remaining zeros of the polynomial.

Quadratic Formula

The quadratic formula is given by:

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

In this case, $a = 1$, $b = -4$, and $c = 8$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(8)}}{2(1)}$$

$$x = \frac{4 \pm \sqrt{16 - 32}}{2}$$

$$x = \frac{4 \pm \sqrt{-16}}{2}$$

Complex Zeros

The quadratic formula gives us complex zeros. Since the discriminant $(b^2 - 4ac)$ is negative, the quadratic factor has complex zeros.

Simplifying Complex Zeros

We can simplify the complex zeros by expressing them in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

$$x = \frac{4 \pm \sqrt{-16}}{2}$$

$$x = \frac{4 \pm 4i}{2}$$

$$x = 2 \pm 2i$$

Remaining Zeros

The remaining zeros of the polynomial function are $2 + 2i$ and $2 - 2i$.

Conclusion

In this article, we used the given zero of a polynomial function to factorize the polynomial and find the remaining zeros. We used polynomial division to divide the polynomial by the given factor and then used the quadratic formula to find the remaining zeros. The remaining zeros were found to be complex numbers.

Final Answer

The remaining zeros of the polynomial function $f(x) = x^3 - 7x^2 + 20x - 24$ are $2 + 2i$ and $2 - 2i$.

Introduction

In our previous article, we discussed how to find the remaining zeros of a polynomial function given that one of its zeros is known. In this article, we will answer some frequently asked questions (FAQs) related to finding remaining zeros of a polynomial function.

Q: What is the first step in finding the remaining zeros of a polynomial function?

A: The first step in finding the remaining zeros of a polynomial function is to factorize the polynomial using the given zero. This can be done using polynomial division or synthetic division.

Q: How do I know if a polynomial is factorable?

A: A polynomial is factorable if it can be expressed as a product of two or more polynomials. In the case of a cubic polynomial, if one of its zeros is known, we can use polynomial division to divide the polynomial by the given factor.

Q: What is the quadratic formula, and how is it used to find remaining zeros?

A: The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. It is given by:

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

In the case of finding remaining zeros of a polynomial function, the quadratic formula is used to find the solutions of the quadratic factor obtained after polynomial division.

Q: What are complex zeros, and how are they expressed?

A: Complex zeros are solutions of a polynomial equation that involve imaginary numbers. They are expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: Can complex zeros be simplified?

A: Yes, complex zeros can be simplified by expressing them in the form $a + bi$.

Q: How do I determine if a polynomial has real or complex zeros?

A: A polynomial has real zeros if the discriminant $(b^2 - 4ac)$ is non-negative. If the discriminant is negative, the polynomial has complex zeros.

Q: Can a polynomial have both real and complex zeros?

A: Yes, a polynomial can have both real and complex zeros. In such cases, the polynomial can be factorized into linear factors, each corresponding to a real or complex zero.

Q: What is the significance of finding remaining zeros of a polynomial function?

A: Finding remaining zeros of a polynomial function is significant in various fields, including mathematics, physics, and engineering. It helps in understanding the behavior of the polynomial function and its applications in real-world problems.

Q: Can you provide examples of polynomial functions with complex zeros?

A: Yes, here are a few examples of polynomial functions with complex zeros:

• f(x) = x^3 - 6x^2 + 11x - 6$ has complex zeros $1 + i$ and $1 - i$.

• f(x) = x^3 - 5x^2 + 6x - 4$ has complex zeros $2 + i$ and $2 - i$.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to finding remaining zeros of a polynomial function. We hope that this article has provided valuable information and insights to readers who are interested in learning more about polynomial functions and their applications.

Final Answer

The remaining zeros of a polynomial function can be found using polynomial division and the quadratic formula. Complex zeros can be expressed in the form $a + bi$ and can be simplified by expressing them in this form.