At Which Root Does The Graph Of F ( X ) = ( X + 4 ) 6 ( X + 7 ) 5 F(x) = (x+4)^6(x+7)^5 F ( X ) = ( X + 4 ) 6 ( X + 7 ) 5 Cross The X X X -axis?A. { -7$}$B. { -4$}$C. 4D. 7

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Introduction

When dealing with polynomial functions, one of the most important tasks is to find the roots of the function, which are the values of the variable that make the function equal to zero. In this article, we will focus on finding the roots of the polynomial function f(x)=(x+4)6(x+7)5f(x) = (x+4)^6(x+7)^5. This function is a product of two binomial factors, each raised to a power. To find the roots of this function, we need to consider the values of xx that make each factor equal to zero.

Understanding the Function

The given function is a product of two binomial factors:

f(x)=(x+4)6(x+7)5f(x) = (x+4)^6(x+7)^5

This function can be expanded using the binomial theorem, but it is not necessary to do so in order to find the roots. Instead, we can use the fact that if a product of two factors is equal to zero, then at least one of the factors must be equal to zero.

Finding the Roots

To find the roots of the function, we need to consider the values of xx that make each factor equal to zero. The first factor is (x+4)6(x+4)^6, and the second factor is (x+7)5(x+7)^5. We can set each factor equal to zero and solve for xx.

First Factor: (x+4)6=0(x+4)^6 = 0

To find the value of xx that makes the first factor equal to zero, we can set (x+4)6=0(x+4)^6 = 0 and solve for xx. Since any number raised to a power of 6 will be zero only if the base is zero, we can set x+4=0x+4 = 0 and solve for xx.

x+4=0x+4 = 0

x=−4x = -4

So, the value of xx that makes the first factor equal to zero is x=−4x = -4.

Second Factor: (x+7)5=0(x+7)^5 = 0

To find the value of xx that makes the second factor equal to zero, we can set (x+7)5=0(x+7)^5 = 0 and solve for xx. Since any number raised to a power of 5 will be zero only if the base is zero, we can set x+7=0x+7 = 0 and solve for xx.

x+7=0x+7 = 0

x=−7x = -7

So, the value of xx that makes the second factor equal to zero is x=−7x = -7.

Conclusion

In conclusion, the roots of the polynomial function f(x)=(x+4)6(x+7)5f(x) = (x+4)^6(x+7)^5 are the values of xx that make each factor equal to zero. We found that the first factor is equal to zero when x=−4x = -4, and the second factor is equal to zero when x=−7x = -7. Therefore, the roots of the function are x=−4x = -4 and x=−7x = -7.

Discussion and Analysis

The roots of a polynomial function are the values of the variable that make the function equal to zero. In this case, we found that the roots of the function f(x)=(x+4)6(x+7)5f(x) = (x+4)^6(x+7)^5 are x=−4x = -4 and x=−7x = -7. This means that the graph of the function crosses the xx-axis at these two points.

The fact that the roots of the function are x=−4x = -4 and x=−7x = -7 has important implications for the behavior of the function. For example, the function will change sign at these two points, and the graph of the function will cross the xx-axis at these points.

Real-World Applications

The concept of roots of a polynomial function has many real-world applications. For example, in physics, the roots of a polynomial function can be used to model the behavior of a system over time. In engineering, the roots of a polynomial function can be used to design and optimize systems such as bridges and buildings.

Conclusion

In conclusion, the roots of a polynomial function are the values of the variable that make the function equal to zero. In this article, we found that the roots of the polynomial function f(x)=(x+4)6(x+7)5f(x) = (x+4)^6(x+7)^5 are x=−4x = -4 and x=−7x = -7. This has important implications for the behavior of the function and has many real-world applications.

Final Answer

The final answer is: −7\boxed{-7}

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form f(x)=anxn+an−1xn−1+…+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0, where an,an−1,…,a1,a0a_n, a_{n-1}, \ldots, a_1, a_0 are constants and nn is a non-negative integer.

Q: What is a root of a polynomial function?

A: A root of a polynomial function is a value of xx that makes the function equal to zero. In other words, if f(x)=0f(x) = 0, then xx is a root of the function.

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

A: To find the roots of a polynomial function, you can use various methods such as factoring, the quadratic formula, or numerical methods. In the case of the function f(x)=(x+4)6(x+7)5f(x) = (x+4)^6(x+7)^5, we can find the roots by setting each factor equal to zero and solving for xx.

Q: What is the difference between a root and a solution?

A: In the context of polynomial functions, the terms "root" and "solution" are often used interchangeably. However, in some cases, a solution may refer to a value of xx that makes the function equal to a specific value, not necessarily zero.

Q: Can a polynomial function have multiple roots?

A: Yes, a polynomial function can have multiple roots. For example, the function f(x)=(x+4)2(x+7)3f(x) = (x+4)^2(x+7)^3 has two roots, x=−4x = -4 and x=−7x = -7, each with a multiplicity of 2 and 3, respectively.

Q: How do I determine the multiplicity of a root?

A: To determine the multiplicity of a root, you can use the fact that if a polynomial function has a root x=rx = r with multiplicity mm, then the function can be written in the form f(x)=(x−r)mg(x)f(x) = (x-r)^m g(x), where g(x)g(x) is a polynomial function.

Q: Can a polynomial function have complex roots?

A: Yes, a polynomial function can have complex roots. For example, the function f(x)=x2+1f(x) = x^2 + 1 has two complex roots, x=ix = i and x=−ix = -i.

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

A: To find the complex roots of a polynomial function, you can use various methods such as the quadratic formula, numerical methods, or the use of complex numbers.

Q: What is the significance of the roots of a polynomial function?

A: The roots of a polynomial function are significant because they determine the behavior of the function. For example, the roots of a function can be used to model the behavior of a system over time, or to design and optimize systems such as bridges and buildings.

Q: Can a polynomial function have no roots?

A: Yes, a polynomial function can have no roots. For example, the function f(x)=x2+1f(x) = x^2 + 1 has no real roots, but it has two complex roots.

Q: How do I determine if a polynomial function has any roots?

A: To determine if a polynomial function has any roots, you can use various methods such as factoring, the quadratic formula, or numerical methods.

Conclusion

In conclusion, the roots of a polynomial function are the values of xx that make the function equal to zero. Finding the roots of a polynomial function is an important task in mathematics and has many real-world applications. We hope that this FAQ article has provided you with a better understanding of the concept of roots and how to find them.