Find The Zeros Of The Function G ( X G(x G ( X ]. G ( X ) = ( X − 9 ) ( X + 4 ) ( X + 2 G(x) = (x-9)(x+4)(x+2 G ( X ) = ( X − 9 ) ( X + 4 ) ( X + 2 ]Write Your Answer As A List Of Values Separated By Commas. X = X = X = □ \square □

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Introduction

In algebra, a polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. The zeros of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will discuss how to find the zeros of a polynomial function, using the function g(x)=(x−9)(x+4)(x+2)g(x) = (x-9)(x+4)(x+2) as an example.

What are Zeros?

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 a zero of the function into the function, the result will be zero. For example, if we have a function f(x)=x2−4f(x) = x^2 - 4, then the zeros of the function are x=2x = 2 and x=−2x = -2, because when we substitute x=2x = 2 or x=−2x = -2 into the function, we get f(2)=f(−2)=0f(2) = f(-2) = 0.

Finding Zeros of a Polynomial Function

To find the zeros of a polynomial function, we need to set the function equal to zero and solve for the variable. In the case of the function g(x)=(x−9)(x+4)(x+2)g(x) = (x-9)(x+4)(x+2), we can set the function equal to zero and solve for xx.

Step 1: Set the Function Equal to Zero

We start by setting the function equal to zero:

g(x)=(x−9)(x+4)(x+2)=0g(x) = (x-9)(x+4)(x+2) = 0

Step 2: Factor the Function

We can factor the function by multiplying out the terms:

g(x)=(x−9)(x+4)(x+2)=x3+x2−38x−72g(x) = (x-9)(x+4)(x+2) = x^3 + x^2 - 38x - 72

Step 3: Solve for x

We can solve for xx by setting each factor equal to zero and solving for xx. In this case, we have three factors:

x−9=0⇒x=9x-9 = 0 \Rightarrow x = 9

x+4=0⇒x=−4x+4 = 0 \Rightarrow x = -4

x+2=0⇒x=−2x+2 = 0 \Rightarrow x = -2

Step 4: List the Zeros

The zeros of the function g(x)g(x) are x=9x = 9, x=−4x = -4, and x=−2x = -2.

Conclusion

In this article, we discussed how to find the zeros of a polynomial function, using the function g(x)=(x−9)(x+4)(x+2)g(x) = (x-9)(x+4)(x+2) as an example. We set the function equal to zero, factored the function, and solved for xx to find the zeros of the function. The zeros of the function g(x)g(x) are x=9x = 9, x=−4x = -4, and x=−2x = -2.

Final Answer

The final answer is: 9,−4,−2\boxed{9, -4, -2}

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Introduction

In our previous article, we discussed how to find the zeros of a polynomial function, using the function g(x)=(x−9)(x+4)(x+2)g(x) = (x-9)(x+4)(x+2) as an example. In this article, we will answer some frequently asked questions (FAQs) about finding zeros of a polynomial function.

Q&A

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

A: A zero and a root of a polynomial function are the same thing. They refer to the values of the variable that make the function equal to zero.

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

A: To find the zeros of a polynomial function with multiple variables, you need to set the function equal to zero and solve for each variable. This can be a complex process, and you may need to use advanced mathematical techniques, such as substitution or elimination.

Q: Can I use a calculator to find the zeros of a polynomial function?

A: Yes, you can use a calculator to find the zeros of a polynomial function. Many calculators have built-in functions for solving polynomial equations, such as the "solve" function on a graphing calculator.

Q: How do I know if a polynomial function has any zeros?

A: To determine if a polynomial function has any zeros, you can use the following methods:

  • Check if the function has any real roots by using the quadratic formula or factoring the function.
  • Use the Rational Root Theorem to determine if the function has any rational roots.
  • Use the Descartes' Rule of Signs to determine if the function has any positive or negative roots.

Q: Can I find the zeros of a polynomial function with complex coefficients?

A: Yes, you can find the zeros of a polynomial function with complex coefficients. However, this can be a complex process, and you may need to use advanced mathematical techniques, such as complex analysis or numerical methods.

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

A: To find the zeros of a polynomial function with a large degree, you can use numerical methods, such as the Newton-Raphson method or the bisection method. These methods can be used to approximate the zeros of the function.

Q: Can I use a computer program to find the zeros of a polynomial function?

A: Yes, you can use a computer program to find the zeros of a polynomial function. Many computer programs, such as MATLAB or Python, have built-in functions for solving polynomial equations.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about finding zeros of a polynomial function. We discussed how to find the zeros of a polynomial function with multiple variables, how to use a calculator to find the zeros of a polynomial function, and how to determine if a polynomial function has any zeros.

Final Answer

The final answer is: Yes,youcanuseacalculatororcomputerprogramtofindthezerosofapolynomialfunction.\boxed{Yes, you can use a calculator or computer program to find the zeros of a polynomial function.}