Select The Correct Answer.Find The Factors Of The Function F ( X ) = X 4 − 5 X 4 − 4 X 2 + 20 X F(x) = X^4 - 5x^4 - 4x^2 + 20x F ( X ) = X 4 − 5 X 4 − 4 X 2 + 20 X .Based On The Factors, Which Statement Is True About The Graph Of The Function?A. The Graph Crosses The X X X -axis At The Point $(2,

Mar 11, 2025 by ADMIN 298 views

**Select the Correct Answer: Factors of the Function and Graph Analysis**

Understanding the Function

The given function is $f(x) = x^4 - 5x^2 - 4x^2 + 20x$ . To find the factors of the function, we need to simplify it first. We can start by combining like terms:

$f(x) = x^4 - 9x^2 + 20x$

Factoring the Function

To factor the function, we can look for common factors. In this case, we can factor out an $x$ from all the terms:

$f(x) = x(x^3 - 9x + 20)$

Now, we need to factor the cubic expression inside the parentheses. We can try to find a rational root using the Rational Root Theorem. The possible rational roots are $\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$ . We can test these values to find a root.

Finding a Root

Let's try to find a root by testing the possible rational roots. We can start by testing the smaller values. If we substitute $x = 1$ into the cubic expression, we get:

$x^3 - 9x + 20 = 1 - 9 + 20 = 12$

Since $x = 1$ is not a root, we can try the next value. If we substitute $x = 2$ into the cubic expression, we get:

$x^3 - 9x + 20 = 8 - 18 + 20 = 10$

Since $x = 2$ is not a root, we can try the next value. If we substitute $x = 4$ into the cubic expression, we get:

$x^3 - 9x + 20 = 64 - 36 + 20 = 48$

Since $x = 4$ is not a root, we can try the next value. If we substitute $x = 5$ into the cubic expression, we get:

$x^3 - 9x + 20 = 125 - 45 + 20 = 100$

Since $x = 5$ is not a root, we can try the next value. If we substitute $x = 10$ into the cubic expression, we get:

$x^3 - 9x + 20 = 1000 - 90 + 20 = 930$

Since $x = 10$ is not a root, we can try the next value. If we substitute $x = 20$ into the cubic expression, we get:

$x^3 - 9x + 20 = 8000 - 180 + 20 = 7840$

Since $x = 20$ is not a root, we can try the next value. If we substitute $x = -1$ into the cubic expression, we get:

$x^3 - 9x + 20 = -1 + 9 + 20 = 28$

Since $x = -1$ is not a root, we can try the next value. If we substitute $x = -2$ into the cubic expression, we get:

$x^3 - 9x + 20 = -8 + 18 + 20 = 30$

Since $x = -2$ is not a root, we can try the next value. If we substitute $x = -4$ into the cubic expression, we get:

$x^3 - 9x + 20 = -64 + 36 + 20 = -8$

Since $x = -4$ is a root, we can factor the cubic expression as:

$x^3 - 9x + 20 = (x + 4)(x^2 - 4x + 5)$

Factoring the Quadratic Expression

Now, we need to factor the quadratic expression inside the parentheses. We can try to find a rational root using the Rational Root Theorem. The possible rational roots are $\pm 1, \pm 5$ . We can test these values to find a root.

Finding a Root

Let's try to find a root by testing the possible rational roots. If we substitute $x = 1$ into the quadratic expression, we get:

$x^2 - 4x + 5 = 1 - 4 + 5 = 2$

Since $x = 1$ is not a root, we can try the next value. If we substitute $x = 5$ into the quadratic expression, we get:

$x^2 - 4x + 5 = 25 - 20 + 5 = 10$

Since $x = 5$ is not a root, we can try the next value. If we substitute $x = -1$ into the quadratic expression, we get:

$x^2 - 4x + 5 = 1 + 4 + 5 = 10$

Since $x = -1$ is not a root, we can try the next value. If we substitute $x = -5$ into the quadratic expression, we get:

$x^2 - 4x + 5 = 25 + 20 + 5 = 50$

Since $x = -5$ is not a root, we can try the next value. We can also try to factor the quadratic expression by finding two numbers whose product is $5$ and whose sum is $-4$ . The numbers are $-5$ and $-1$ . Therefore, we can factor the quadratic expression as:

$x^2 - 4x + 5 = (x - 5)(x - 1)$

Factoring the Function

Now, we can factor the function as:

$f(x) = x(x + 4)(x - 5)(x - 1)$

Analyzing the Graph

The graph of the function is a polynomial function of degree $4$ . Since the function has four real roots, the graph will have four $x$ -intercepts. The roots of the function are $x = 0, x = -4, x = 5, x = 1$ . Therefore, the graph will cross the $x$ -axis at these points.

Conclusion

Based on the factors of the function, we can conclude that the graph of the function crosses the $x$ -axis at the points $(0, 0), (-4, 0), (5, 0), (1, 0)$ . Therefore, the correct answer is:

A. The graph crosses the $x$ -axis at the points $(0, 0), (-4, 0), (5, 0), (1, 0)$ .

Discussion

The graph of a polynomial function can be analyzed by finding its roots and factoring the function. In this case, we found the roots of the function to be $x = 0, x = -4, x = 5, x = 1$ . We also factored the function as $f(x) = x(x + 4)(x - 5)(x - 1)$ . The graph of the function will cross the $x$ -axis at these points.

Q: What is the degree of the polynomial function?

A: The degree of the polynomial function is $4$ . This means that the highest power of the variable $x$ in the function is $4$ .

Q: How many real roots does the function have?

A: The function has four real roots, which are $x = 0, x = -4, x = 5, x = 1$ .

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

A: The x-intercept of the graph is the point where the graph crosses the x-axis. In this case, the graph crosses the x-axis at the points $(0, 0), (-4, 0), (5, 0), (1, 0)$ .

Q: How can we analyze the graph of a polynomial function?

A: We can analyze the graph of a polynomial function by finding its roots and factoring the function. By factoring the function, we can identify the x-intercepts of the graph.

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

A: The roots of a polynomial function are the values of the variable that make the function equal to zero. In this case, the roots of the function are $x = 0, x = -4, x = 5, x = 1$ . These roots are significant because they represent the x-intercepts of the graph.

Q: How can we use the factors of a polynomial function to analyze its graph?

A: We can use the factors of a polynomial function to analyze its graph by identifying the x-intercepts of the graph. By factoring the function, we can identify the values of the variable that make the function equal to zero, which are the x-intercepts of the graph.

Q: What is the relationship between the factors of a polynomial function and its graph?

A: The factors of a polynomial function are related to its graph in that they represent the x-intercepts of the graph. By factoring the function, we can identify the x-intercepts of the graph, which are the points where the graph crosses the x-axis.

Q: Can we use the factors of a polynomial function to determine the behavior of its graph?

A: Yes, we can use the factors of a polynomial function to determine the behavior of its graph. By analyzing the factors of the function, we can identify the x-intercepts of the graph, which can help us determine the behavior of the graph.

Q: How can we use the graph of a polynomial function to analyze its factors?

A: We can use the graph of a polynomial function to analyze its factors by identifying the x-intercepts of the graph. By analyzing the x-intercepts of the graph, we can identify the values of the variable that make the function equal to zero, which are the factors of the function.

Q: What is the significance of the graph of a polynomial function in relation to its factors?

A: The graph of a polynomial function is significant in relation to its factors because it represents the x-intercepts of the graph, which are the values of the variable that make the function equal to zero. By analyzing the graph of the function, we can identify the factors of the function, which can help us understand the behavior of the function.

Q: Can we use the graph of a polynomial function to determine the degree of the function?

A: Yes, we can use the graph of a polynomial function to determine the degree of the function. By analyzing the graph of the function, we can identify the highest power of the variable that makes the function equal to zero, which is the degree of the function.

Q: How can we use the graph of a polynomial function to analyze its behavior?

A: We can use the graph of a polynomial function to analyze its behavior by identifying the x-intercepts of the graph, which represent the values of the variable that make the function equal to zero. By analyzing the x-intercepts of the graph, we can determine the behavior of the function.

Q: What is the relationship between the graph of a polynomial function and its behavior?

A: The graph of a polynomial function is related to its behavior in that it represents the x-intercepts of the graph, which are the values of the variable that make the function equal to zero. By analyzing the graph of the function, we can determine the behavior of the function.

Q: Can we use the graph of a polynomial function to determine the roots of the function?

A: Yes, we can use the graph of a polynomial function to determine the roots of the function. By analyzing the x-intercepts of the graph, we can identify the values of the variable that make the function equal to zero, which are the roots of the function.

Q: How can we use the graph of a polynomial function to analyze its factors?

A: We can use the graph of a polynomial function to analyze its factors by identifying the x-intercepts of the graph, which represent the values of the variable that make the function equal to zero. By analyzing the x-intercepts of the graph, we can determine the factors of the function.