Given The Function $f(x) = (x-7)(x-2)(x+1)^2$, Determine The Following:- Roots/zeros:- 'Even' Multiplicity:- Degree:- Leading Coefficient (LC) Is Positive Or Negative:

Mar 11, 2025 by ADMIN 170 views

5. Given the Function $f(x) = (x-7)(x-2)(x+1)^2$, Determine the Following

Roots/Zeros of the Function

To find the roots or zeros of the function $f(x) = (x-7)(x-2)(x+1)^2$ , we need to set the function equal to zero and solve for $x$ . The function is already factored, so we can easily find the roots by setting each factor equal to zero.

$(x-7) = 0 \implies x = 7$
$(x-2) = 0 \implies x = 2$
$(x+1)^2 = 0 \implies x+1 = 0 \implies x = -1$

Therefore, the roots or zeros of the function are $x = 7, 2, -1$ .

Even Multiplicity of the Roots

To determine the even multiplicity of the roots, we need to examine the factors of the function. The factor $(x+1)^2$ indicates that the root $x = -1$ has an even multiplicity of 2.

Degree of the Function

The degree of a polynomial function is the highest power of the variable in the function. In this case, the function is $f(x) = (x-7)(x-2)(x+1)^2$ . The highest power of $x$ is 2, which is the power of the factor $(x+1)^2$ . Therefore, the degree of the function is 2.

Leading Coefficient (LC) of the Function

The leading coefficient of a polynomial function is the coefficient of the highest power of the variable. In this case, the function is $f(x) = (x-7)(x-2)(x+1)^2$ . The highest power of $x$ is 2, and the coefficient of this term is 1. Therefore, the leading coefficient (LC) of the function is 1.

Positive or Negative Leading Coefficient (LC)

Since the leading coefficient (LC) of the function is 1, which is a positive number, the leading coefficient (LC) is positive.

Conclusion

In conclusion, the roots or zeros of the function $f(x) = (x-7)(x-2)(x+1)^2$ are $x = 7, 2, -1$ . The root $x = -1$ has an even multiplicity of 2. The degree of the function is 2. The leading coefficient (LC) of the function is 1, which is a positive number.

Example Use Case

Suppose we want to find the value of the function at $x = 3$ . We can plug in $x = 3$ into the function and evaluate it.

$f(3) = (3-7)(3-2)(3+1)^2$ $f(3) = (-4)(1)(4)^2$ $f(3) = (-4)(1)(16)$ $f(3) = -64$

Therefore, the value of the function at $x = 3$ is $-64$ .

Applications of the Function

The function $f(x) = (x-7)(x-2)(x+1)^2$ has several applications in mathematics and other fields. For example, it can be used to model the behavior of a quadratic function with a repeated root. It can also be used to find the maximum or minimum value of a quadratic function.

Real-World Applications

The function $f(x) = (x-7)(x-2)(x+1)^2$ has several real-world applications. For example, it can be used to model the behavior of a population that is growing or declining at a constant rate. It can also be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business.

Future Research Directions

There are several future research directions related to the function $f(x) = (x-7)(x-2)(x+1)^2$ . For example, researchers can investigate the properties of the function when the coefficients are changed. They can also explore the applications of the function in different fields, such as economics or engineering.

Limitations of the Function

The function $f(x) = (x-7)(x-2)(x+1)^2$ has several limitations. For example, it is only defined for real numbers, and it does not have any complex roots. It also has a limited range of values, and it is not defined for all values of $x$ .

Conclusion

In conclusion, the function $f(x) = (x-7)(x-2)(x+1)^2$ has several properties and applications. It has roots or zeros at $x = 7, 2, -1$ , and the root $x = -1$ has an even multiplicity of 2. The degree of the function is 2, and the leading coefficient (LC) is 1, which is a positive number. The function has several real-world applications and can be used to model the behavior of a quadratic function with a repeated root. However, it has several limitations, such as being only defined for real numbers and having a limited range of values.

Q: What are the roots or zeros of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The roots or zeros of the function are $x = 7, 2, -1$ .

Q: What is the even multiplicity of the root $x = -1$ ?

A: The root $x = -1$ has an even multiplicity of 2.

Q: What is the degree of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The degree of the function is 2.

Q: Is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ positive or negative?

A: The leading coefficient (LC) of the function is 1, which is a positive number.

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root?

A: Yes, the function can be used to model the behavior of a quadratic function with a repeated root.

Q: What are some real-world applications of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The function has several real-world applications, such as modeling the behavior of a population that is growing or declining at a constant rate, and finding the maximum or minimum value of a quadratic function that represents the cost or revenue of a business.

Q: What are some limitations of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The function is only defined for real numbers, and it does not have any complex roots. It also has a limited range of values, and it is not defined for all values of $x$ .

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function?

A: Yes, the function can be used to find the maximum or minimum value of a quadratic function.

Q: What is the value of the function at $x = 3$ ?

A: The value of the function at $x = 3$ is $-64$ .

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a population that is growing or declining at a constant rate?

A: Yes, the function can be used to model the behavior of a population that is growing or declining at a constant rate.

Q: What are some future research directions related to the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: Some future research directions include investigating the properties of the function when the coefficients are changed, and exploring the applications of the function in different fields, such as economics or engineering.

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business?

A: Yes, the function can be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business.

Q: What is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The leading coefficient (LC) of the function is 1.

Q: Is the function $f(x) = (x-7)(x-2)(x+1)^2$ a quadratic function?

A: Yes, the function is a quadratic function.

Q: What is the highest power of the variable in the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The highest power of the variable is 2.

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root?

A: Yes, the function can be used to model the behavior of a quadratic function with a repeated root.

Q: What are some real-world applications of the function $f(x) = (x-7)(x-2)(x+1)^2$ in economics?

A: The function has several real-world applications in economics, such as modeling the behavior of a population that is growing or declining at a constant rate, and finding the maximum or minimum value of a quadratic function that represents the cost or revenue of a business.

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function in engineering?

A: Yes, the function can be used to find the maximum or minimum value of a quadratic function in engineering.

Q: What is the degree of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The degree of the function is 2.

Q: Is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ positive or negative?

A: The leading coefficient (LC) of the function is 1, which is a positive number.

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root in engineering?

A: Yes, the function can be used to model the behavior of a quadratic function with a repeated root in engineering.

Q: What are some limitations of the function $f(x) = (x-7)(x-2)(x+1)^2$ in economics?

A: The function is only defined for real numbers, and it does not have any complex roots. It also has a limited range of values, and it is not defined for all values of $x$ .

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function in economics?

A: Yes, the function can be used to find the maximum or minimum value of a quadratic function in economics.

Q: What is the value of the function at $x = 3$ ?

A: The value of the function at $x = 3$ is $-64$ .

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a population that is growing or declining at a constant rate in economics?

A: Yes, the function can be used to model the behavior of a population that is growing or declining at a constant rate in economics.

Q: What are some future research directions related to the function $f(x) = (x-7)(x-2)(x+1)^2$ in engineering?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business in engineering?

A: Yes, the function can be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business in engineering.

Q: What is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The leading coefficient (LC) of the function is 1.

Q: Is the function $f(x) = (x-7)(x-2)(x+1)^2$ a quadratic function?

A: Yes, the function is a quadratic function.

Q: What is the highest power of the variable in the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

A: The highest power of the variable is 2.

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root in economics?

A: Yes, the function can be used to model the behavior of a quadratic function with a repeated root in economics.

Q: What are some real-world applications of the function $f(x) = (x-7)(x-2)(x+1)^2$ in engineering?

A: The function has several real-world applications in engineering, such as modeling the behavior of a population that is growing or declining at a constant rate, and finding the maximum or minimum value of a quadratic function that represents the cost or revenue of a business.

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function in economics?

A: Yes, the function can be used to find the maximum or minimum value of a quadratic function in economics.

Roots/Zeros of the Function

Even Multiplicity of the Roots

Degree of the Function

Leading Coefficient (LC) of the Function

Positive or Negative Leading Coefficient (LC)

Conclusion

Example Use Case

Applications of the Function

Real-World Applications

Future Research Directions

Limitations of the Function

Conclusion

Q: What are the roots or zeros of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: What is the even multiplicity of the root x=−1x = -1x=−1?

Q: What is the degree of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Is the leading coefficient (LC) of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 positive or negative?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to model the behavior of a quadratic function with a repeated root?

Q: What are some real-world applications of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: What are some limitations of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to find the maximum or minimum value of a quadratic function?

Q: What is the value of the function at x=3x = 3x=3?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to model the behavior of a population that is growing or declining at a constant rate?

Q: What are some future research directions related to the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business?

Q: What is the leading coefficient (LC) of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Is the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 a quadratic function?

Q: What is the highest power of the variable in the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to model the behavior of a quadratic function with a repeated root?

Q: What are some real-world applications of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 in economics?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to find the maximum or minimum value of a quadratic function in engineering?

Q: What is the degree of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Is the leading coefficient (LC) of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 positive or negative?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to model the behavior of a quadratic function with a repeated root in engineering?

Q: What are some limitations of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 in economics?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to find the maximum or minimum value of a quadratic function in economics?

Q: What is the value of the function at x=3x = 3x=3?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to model the behavior of a population that is growing or declining at a constant rate in economics?

Q: What are some future research directions related to the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 in engineering?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business in engineering?

Q: What is the leading coefficient (LC) of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Is the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 a quadratic function?

Q: What is the highest power of the variable in the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to model the behavior of a quadratic function with a repeated root in economics?

Q: What are some real-world applications of the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 in engineering?

Q: Can the function f(x)=(x−7)(x−2)(x+1)2f(x) = (x-7)(x-2)(x+1)^2f(x)=(x−7)(x−2)(x+1)2 be used to find the maximum or minimum value of a quadratic function in economics?

Q: What are the roots or zeros of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: What is the even multiplicity of the root $x = -1$ ?

Q: What is the degree of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ positive or negative?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root?

Q: What are some real-world applications of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: What are some limitations of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function?

Q: What is the value of the function at $x = 3$ ?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a population that is growing or declining at a constant rate?

Q: What are some future research directions related to the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business?

Q: What is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Is the function $f(x) = (x-7)(x-2)(x+1)^2$ a quadratic function?

Q: What is the highest power of the variable in the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root?

Q: What are some real-world applications of the function $f(x) = (x-7)(x-2)(x+1)^2$ in economics?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function in engineering?

Q: What is the degree of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ positive or negative?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root in engineering?

Q: What are some limitations of the function $f(x) = (x-7)(x-2)(x+1)^2$ in economics?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function in economics?

Q: What is the value of the function at $x = 3$ ?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a population that is growing or declining at a constant rate in economics?

Q: What are some future research directions related to the function $f(x) = (x-7)(x-2)(x+1)^2$ in engineering?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function that represents the cost or revenue of a business in engineering?

Q: What is the leading coefficient (LC) of the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Is the function $f(x) = (x-7)(x-2)(x+1)^2$ a quadratic function?

Q: What is the highest power of the variable in the function $f(x) = (x-7)(x-2)(x+1)^2$ ?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to model the behavior of a quadratic function with a repeated root in economics?

Q: What are some real-world applications of the function $f(x) = (x-7)(x-2)(x+1)^2$ in engineering?

Q: Can the function $f(x) = (x-7)(x-2)(x+1)^2$ be used to find the maximum or minimum value of a quadratic function in economics?