Find The Root(s) Of $f(x)=(x-6)^2(x+2)^2$.A. -6 With Multiplicity 1 B. -6 With Multiplicity 2 C. 6 With Multiplicity 1 D. 6 With Multiplicity 2 E. -2 With Multiplicity 1 F. -2 With Multiplicity 2 G. 2 With Multiplicity 1 H. 2 With
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Introduction
In algebra, finding the roots of a polynomial function is a crucial concept that helps us understand the behavior of the function. A root of a polynomial function is a value of x that makes the function equal to zero. In this article, we will focus on finding the roots of the polynomial function f(x) = (x-6)2(x+2)2.
Understanding the Function
The given function is a quadratic function that has been squared. This means that it has two roots, but with a multiplicity of 2. To find the roots, we need to set the function equal to zero and solve for x.
Setting the Function Equal to Zero
To find the roots of the function, we need to set it equal to zero and solve for x. We can do this by setting each factor equal to zero and solving for x.
(x-6)^2 = 0 (x+2)^2 = 0
Solving for x
Now that we have set each factor equal to zero, we can solve for x.
x-6 = 0 x+2 = 0
Finding the Roots
Now that we have solved for x, we can find the roots of the function.
x = 6 x = -2
Multiplicity of the Roots
Since the function has been squared, the roots have a multiplicity of 2. This means that the root x = 6 has a multiplicity of 2, and the root x = -2 has a multiplicity of 2.
Conclusion
In conclusion, the roots of the polynomial function f(x) = (x-6)2(x+2)2 are x = 6 with multiplicity 2 and x = -2 with multiplicity 2.
Answer
The correct answer is:
- D. 6 with multiplicity 2
- F. -2 with multiplicity 2
Final Answer
The final answer is D and F.
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Introduction
In our previous article, we discussed how to find the roots of a polynomial function f(x) = (x-6)2(x+2)2. In this article, we will answer some frequently asked questions related to finding the roots of a polynomial function.
Q&A
Q: What is the difference between a root and a zero of a polynomial function?
A: A root of a polynomial function is a value of x that makes the function equal to zero. A zero of a polynomial function is also a value of x that makes the function equal to zero, but it is a more general term that can be used for any value of x that makes the function equal to zero, not just the roots.
Q: How do I find the roots of a polynomial function?
A: To find the roots of a polynomial function, you need to set the function equal to zero and solve for x. You can do this by setting each factor equal to zero and solving for x.
Q: What is the multiplicity of a root?
A: The multiplicity of a root is the number of times that the root appears in the factorization of the polynomial function. For example, if a root has a multiplicity of 2, it means that the root appears twice in the factorization of the polynomial function.
Q: How do I determine the multiplicity of a root?
A: To determine the multiplicity of a root, you need to look at the factorization of the polynomial function. If a root appears once in the factorization, it has a multiplicity of 1. If a root appears twice in the factorization, it has a multiplicity of 2, and so on.
Q: Can a polynomial function have more than one root?
A: Yes, a polynomial function can have more than one root. In fact, a polynomial function can have any number of roots, depending on its degree.
Q: How do I find the roots of a polynomial function with multiple roots?
A: To find the roots of a polynomial function with multiple roots, you need to set the function equal to zero and solve for x. You can do this by setting each factor equal to zero and solving for x.
Q: What is the difference between a quadratic function and a polynomial function?
A: A quadratic function is a polynomial function of degree 2, while a polynomial function can be of any degree.
Q: Can a polynomial function have a root that is not an integer?
A: Yes, a polynomial function can have a root that is not an integer. In fact, a polynomial function can have any type of root, including rational roots, irrational roots, and complex roots.
Conclusion
In conclusion, finding the roots of a polynomial function is an important concept in algebra. By understanding the concept of roots and how to find them, you can solve a wide range of problems in mathematics and other fields.
Final Answer
The final answer is that finding the roots of a polynomial function is a crucial concept in algebra that can be used to solve a wide range of problems.