What Are The Zeros Of The Polynomial Function? F ( X ) = X 2 − 4 X − 60 F(x)=x^2-4x-60 F ( X ) = X 2 − 4 X − 60 Enter Your Answers In The Boxes.The Zeros Of F ( X F(x F ( X ] Are □ \square □ And □ \square □ .

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Introduction

In mathematics, a polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The zeros of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will explore the concept of zeros of a polynomial function and find the zeros of the given polynomial function f(x)=x2−4x−60f(x) = x^2 - 4x - 60.

What are Zeros of a Polynomial Function?

The zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we substitute the zero of a polynomial function into the function, the result will be zero. The zeros of a polynomial function are also known as the roots of the polynomial.

How to Find the Zeros of a Polynomial Function?

There are several methods to find the zeros of a polynomial function, including factoring, the quadratic formula, and graphing. In this article, we will use the quadratic formula to find the zeros of the given polynomial function.

The Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0. The quadratic formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Finding the Zeros of the Polynomial Function

To find the zeros of the polynomial function f(x)=x2−4x−60f(x) = x^2 - 4x - 60, we will use the quadratic formula. The coefficients of the polynomial function are a=1a = 1, b=−4b = -4, and c=−60c = -60. Substituting these values into the quadratic formula, we get:

x=−(−4)±(−4)2−4(1)(−60)2(1)x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-60)}}{2(1)}

Simplifying the expression, we get:

x=4±16+2402x = \frac{4 \pm \sqrt{16 + 240}}{2}

x=4±2562x = \frac{4 \pm \sqrt{256}}{2}

x=4±162x = \frac{4 \pm 16}{2}

Therefore, the zeros of the polynomial function f(x)=x2−4x−60f(x) = x^2 - 4x - 60 are:

x=4+162=10x = \frac{4 + 16}{2} = 10

x=4−162=−6x = \frac{4 - 16}{2} = -6

Conclusion

In this article, we have explored the concept of zeros of a polynomial function and found the zeros of the given polynomial function f(x)=x2−4x−60f(x) = x^2 - 4x - 60. The zeros of a polynomial function are the values of the variable that make the function equal to zero. We have used the quadratic formula to find the zeros of the polynomial function. The zeros of the polynomial function are x=10x = 10 and x=−6x = -6.

Applications of Zeros of a Polynomial Function

The zeros of a polynomial function have several applications in mathematics and science. Some of the applications of zeros of a polynomial function include:

  • Graphing: The zeros of a polynomial function are the x-intercepts of the graph of the function.
  • Solving Equations: The zeros of a polynomial function can be used to solve equations of the form f(x)=0f(x) = 0.
  • Optimization: The zeros of a polynomial function can be used to find the maximum or minimum value of a function.
  • Science: The zeros of a polynomial function can be used to model real-world phenomena, such as the motion of an object under the influence of gravity.

Real-World Examples of Zeros of a Polynomial Function

Some real-world examples of zeros of a polynomial function include:

  • Projectile Motion: The zeros of a polynomial function can be used to model the motion of a projectile under the influence of gravity.
  • Population Growth: The zeros of a polynomial function can be used to model the growth of a population over time.
  • Economics: The zeros of a polynomial function can be used to model the behavior of economic systems, such as the supply and demand of a product.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about zeros of a polynomial function.

Q: What is the difference between a zero and a root of a polynomial function?

A: The terms "zero" and "root" are often used interchangeably to refer to the values of the variable that make the function equal to zero. However, some mathematicians make a distinction between the two terms. A zero is a value of the variable that makes the function equal to zero, while a root is a value of the variable that makes the function equal to zero and is also a solution to the equation.

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

A: There are several methods to find the zeros of a polynomial function, including factoring, the quadratic formula, and graphing. The method you choose will depend on the complexity of the polynomial function and the tools you have available.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0. The quadratic formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to find the zeros of a polynomial function?

A: To use the quadratic formula to find the zeros of a polynomial function, you need to identify the coefficients of the polynomial function, which are aa, bb, and cc. Then, you can substitute these values into the quadratic formula and simplify the expression to find the zeros of the polynomial function.

Q: What are some common mistakes to avoid when finding the zeros of a polynomial function?

A: Some common mistakes to avoid when finding the zeros of a polynomial function include:

  • Not checking the domain of the function: Make sure that the values you are using to find the zeros of the polynomial function are in the domain of the function.
  • Not simplifying the expression: Make sure to simplify the expression you get from the quadratic formula to find the zeros of the polynomial function.
  • Not checking for extraneous solutions: Make sure to check for extraneous solutions, which are solutions that are not actually zeros of the polynomial function.

Q: What are some real-world applications of zeros of a polynomial function?

A: Some real-world applications of zeros of a polynomial function include:

  • Graphing: The zeros of a polynomial function are the x-intercepts of the graph of the function.
  • Solving Equations: The zeros of a polynomial function can be used to solve equations of the form f(x)=0f(x) = 0.
  • Optimization: The zeros of a polynomial function can be used to find the maximum or minimum value of a function.
  • Science: The zeros of a polynomial function can be used to model real-world phenomena, such as the motion of an object under the influence of gravity.

Q: How do I use technology to find the zeros of a polynomial function?

A: There are several ways to use technology to find the zeros of a polynomial function, including:

  • Graphing calculators: Graphing calculators can be used to graph the function and find the x-intercepts, which are the zeros of the polynomial function.
  • Computer algebra systems: Computer algebra systems, such as Mathematica or Maple, can be used to find the zeros of a polynomial function.
  • Online tools: There are several online tools, such as Wolfram Alpha or Symbolab, that can be used to find the zeros of a polynomial function.

Conclusion

In conclusion, the zeros of a polynomial function are the values of the variable that make the function equal to zero. We have answered some of the most frequently asked questions about zeros of a polynomial function, including how to find the zeros of a polynomial function, what is the quadratic formula, and how to use technology to find the zeros of a polynomial function.