For The Polynomial Function F ( X ) = 7 ( X 2 + 4 ) 2 ( X − 4 ) 3 F(x)=7\left(x^2+4\right)^2(x-4)^3 F ( X ) = 7 ( X 2 + 4 ) 2 ( X − 4 ) 3 , Answer The Following Questions:(a) List Each Real Zero And Its Multiplicity.(b) Determine Whether The Graph Crosses Or Touches The X-axis At Each X-intercept.(c) Determine The

Introduction

Polynomial functions are a fundamental concept in algebra, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the analysis of the polynomial function $f(x)=7\left(x^2+4\right)^2(x-4)^3$ and answer three essential questions related to its real zeros, x-intercepts, and graph behavior.

Question (a): List Each Real Zero and Its Multiplicity

To find the real zeros of the polynomial function, we need to set $f(x)$ equal to zero and solve for $x$. The given function can be rewritten as:

$f(x) = 7(x^2 + 4)^2(x - 4)^3$

We can start by factoring the expression:

$f(x) = 7(x^2 + 4)^2(x - 4)^3$

Now, we can set each factor equal to zero and solve for $x$:

1. $(x^2 + 4)^2 = 0$
2. $(x - 4)^3 = 0$

Solving the first equation, we get:

$x^2 + 4 = 0$

$x^2 = -4$

$x = \pm 2i$

Since $x$ is not a real number, this solution is discarded.

Solving the second equation, we get:

$x - 4 = 0$

$x = 4$

Therefore, the only real zero of the polynomial function is $x = 4$.

However, we need to consider the multiplicity of the zero. The multiplicity of a zero is the number of times the factor $(x - 4)$ appears in the polynomial function. In this case, the factor $(x - 4)$ appears three times, so the multiplicity of the zero $x = 4$ is 3.

Conclusion

The real zero of the polynomial function $f(x)=7\left(x^2+4\right)^2(x-4)^3$ is $x = 4$, and its multiplicity is 3.

Question (b): Determine Whether the Graph Crosses or Touches the X-Axis at Each X-Intercept

To determine whether the graph crosses or touches the x-axis at each x-intercept, we need to examine the behavior of the polynomial function near the x-intercept.

In this case, the only x-intercept is $x = 4$. To determine whether the graph crosses or touches the x-axis at this point, we can examine the behavior of the polynomial function as $x$ approaches 4.

As $x$ approaches 4 from the left, the polynomial function approaches negative infinity. As $x$ approaches 4 from the right, the polynomial function approaches positive infinity.

Therefore, the graph of the polynomial function touches the x-axis at $x = 4$.

Conclusion

The graph of the polynomial function $f(x)=7\left(x^2+4\right)^2(x-4)^3$ touches the x-axis at $x = 4$.

Question (c): Determine the Behavior of the Graph Near the X-Intercept

To determine the behavior of the graph near the x-intercept, we can examine the behavior of the polynomial function as $x$ approaches the x-intercept.

In this case, the x-intercept is $x = 4$. As $x$ approaches 4 from the left, the polynomial function approaches negative infinity. As $x$ approaches 4 from the right, the polynomial function approaches positive infinity.

Therefore, the graph of the polynomial function has a vertical tangent at $x = 4$.

Conclusion

The graph of the polynomial function $f(x)=7\left(x^2+4\right)^2(x-4)^3$ has a vertical tangent at $x = 4$.

Conclusion

In conclusion, the polynomial function $f(x)=7\left(x^2+4\right)^2(x-4)^3$ has a real zero at $x = 4$ with a multiplicity of 3. The graph of the polynomial function touches the x-axis at $x = 4$ and has a vertical tangent at this point.

Key Takeaways

• The real zero of the polynomial function $f(x)=7\left(x^2+4\right)^2(x-4)^3$ is $x = 4$, and its multiplicity is 3.
• The graph of the polynomial function touches the x-axis at $x = 4$.
• The graph of the polynomial function has a vertical tangent at $x = 4$.

Real-World Applications

Polynomial functions have numerous real-world applications in various fields, including:

• Physics: Polynomial functions are used to model the motion of objects under the influence of forces.
• Engineering: Polynomial functions are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Economics: Polynomial functions are used to model economic systems and make predictions about future economic trends.

Conclusion

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Q: What is a polynomial function?

A polynomial function is a function that can be written in the form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $n$ is a non-negative integer.

Q: What are the different types of polynomial functions?

There are several types of polynomial functions, including:

• Linear polynomial functions: These are polynomial functions of degree 1, which can be written in the form $f(x) = ax + b$.
• Quadratic polynomial functions: These are polynomial functions of degree 2, which can be written in the form $f(x) = ax^2 + bx + c$.
• Cubic polynomial functions: These are polynomial functions of degree 3, which can be written in the form $f(x) = ax^3 + bx^2 + cx + d$.
• Quartic polynomial functions: These are polynomial functions of degree 4, which can be written in the form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$.

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

To find the real zeros of a polynomial function, you can use the following steps:

1. Set the polynomial function equal to zero: $f(x) = 0$
2. Factor the polynomial function: $f(x) = (x - r_1)(x - r_2)\ldots(x - r_n)$
3. Solve for $x$: $x = r_1, r_2, \ldots, r_n$

Q: What is the multiplicity of a zero?

The multiplicity of a zero is the number of times the factor $(x - r)$ appears in the polynomial function.

Q: How do I determine whether the graph crosses or touches the x-axis at each x-intercept?

To determine whether the graph crosses or touches the x-axis at each x-intercept, you can examine the behavior of the polynomial function near the x-intercept.

Q: What is the behavior of the graph near the x-intercept?

The behavior of the graph near the x-intercept can be determined by examining the behavior of the polynomial function as $x$ approaches the x-intercept.

Q: How do I determine the behavior of the graph near the x-intercept?

To determine the behavior of the graph near the x-intercept, you can use the following steps:

1. Examine the behavior of the polynomial function as $x$ approaches the x-intercept from the left.
2. Examine the behavior of the polynomial function as $x$ approaches the x-intercept from the right.
3. Determine whether the graph crosses or touches the x-axis at the x-intercept.

Q: What is the significance of the multiplicity of a zero?

The multiplicity of a zero is significant because it determines the behavior of the graph near the x-intercept.

Q: How do I use polynomial functions in real-world applications?

Polynomial functions have numerous real-world applications in various fields, including:

• Physics: Polynomial functions are used to model the motion of objects under the influence of forces.
• Engineering: Polynomial functions are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Economics: Polynomial functions are used to model economic systems and make predictions about future economic trends.

Conclusion

In conclusion, polynomial functions are a fundamental concept in algebra, and understanding their properties is crucial for solving various mathematical problems. The Q&A section has provided valuable insights into the different types of polynomial functions, how to find the real zeros of a polynomial function, and how to determine the behavior of the graph near the x-intercept.