Evaluating the Polynomial at the Given Point

Oct 13, 2025 by magicalceylontours.com 18 views

Introduction

When dealing with polynomials, understanding their properties and behavior is crucial. In this case, we are given a polynomial with specific $x$ -intercepts and a point through which it passes. Our goal is to determine which of the given functions could represent this graph.

Understanding the $x$ -Intercepts

The $x$ -intercepts of a polynomial are the values of $x$ where the graph intersects the $x$ -axis. In this case, we are given that the $x$ -intercepts are 3, 0, and -1. This means that the polynomial has factors of $(x - 3)$ , $(x - 0)$ , and $(x + 1)$ .

Factoring the Polynomial

Using the given $x$ -intercepts, we can write the polynomial as a product of its factors:

P(x) = (x - 3)(x - 0)(x + 1)

Simplifying this expression, we get:

P(x) = (x - 3)x(x + 1)

Expanding the Polynomial

To determine which of the given functions could represent this graph, we need to expand the polynomial:

P(x) = (x - 3)x(x + 1)

P(x) = x(x^2 - 3x + x - 3)

P(x) = x(x^2 - 2x - 3)

P(x) = x^3 - 2x^2 - 3x

Evaluating the Polynomial at the Given Point

We are given that the polynomial passes through the point $(1, -8)$ . To determine if this is true, we can evaluate the polynomial at $x = 1$ :

P(1) = (1)^3 - 2(1)^2 - 3(1)

P(1) = 1 - 2 - 3

P(1) = -4

However, we are given that the polynomial passes through the point $(1, -8)$ . This means that the value of the polynomial at $x = 1$ should be $-8$ , not $-4$ .

Comparing the Polynomial to the Given Functions

Let's compare the expanded polynomial to the given functions:

A. $x^3 - 2x^2 - 3x$

B. $x^2 - 2x - 3$

C. $2x^2 - 4x - 6$

D. (no function given)

The expanded polynomial matches function A.

Conclusion

Based on the given $x$ -intercepts and the point through which the polynomial passes, we can conclude that the polynomial is represented by function A: $x^3 - 2x^2 - 3x$ .

Discussion

This problem requires a deep understanding of polynomial properties and behavior. The key concept here is that the polynomial has specific $x$ -intercepts, which can be used to write the polynomial as a product of its factors. By expanding the polynomial and evaluating it at the given point, we can determine which of the given functions could represent this graph.

Additional Tips and Tricks

When dealing with polynomials, it's essential to understand the properties of the graph, including the $x$ -intercepts and the behavior of the polynomial.
The factors of a polynomial can be used to write the polynomial as a product of its factors.
Expanding a polynomial can help determine its behavior and properties.
Evaluating a polynomial at a given point can help determine if the polynomial passes through that point.

Final Answer

The final answer is: $\boxed{A}$

Introduction

Understanding the $x$ -Intercepts

Factoring the Polynomial

Using the given $x$ -intercepts, we can write the polynomial as a product of its factors:

P(x) = (x - 3)(x - 0)(x + 1)

Simplifying this expression, we get:

P(x) = (x - 3)x(x + 1)

Expanding the Polynomial

To determine which of the given functions could represent this graph, we need to expand the polynomial:

P(x) = (x - 3)x(x + 1)

P(x) = x(x^2 - 3x + x - 3)

P(x) = x(x^2 - 2x - 3)

P(x) = x^3 - 2x^2 - 3x

Evaluating the Polynomial at the Given Point

We are given that the polynomial passes through the point $(1, -8)$ . To determine if this is true, we can evaluate the polynomial at $x = 1$ :

P(1) = (1)^3 - 2(1)^2 - 3(1)

P(1) = 1 - 2 - 3

P(1) = -4

However, we are given that the polynomial passes through the point $(1, -8)$ . This means that the value of the polynomial at $x = 1$ should be $-8$ , not $-4$ .

Comparing the Polynomial to the Given Functions

Let's compare the expanded polynomial to the given functions:

A. $x^3 - 2x^2 - 3x$

B. $x^2 - 2x - 3$

C. $2x^2 - 4x - 6$

D. (no function given)

The expanded polynomial matches function A.

Conclusion

Based on the given $x$ -intercepts and the point through which the polynomial passes, we can conclude that the polynomial is represented by function A: $x^3 - 2x^2 - 3x$ .

Q&A

Q: What are the $x$ -intercepts of the polynomial?

A: The $x$ -intercepts of the polynomial are 3, 0, and -1.

Q: How can we write the polynomial as a product of its factors?

A: We can write the polynomial as a product of its factors by using the given $x$ -intercepts. In this case, we can write the polynomial as $(x - 3)(x - 0)(x + 1)$ .

Q: How can we expand the polynomial?

A: We can expand the polynomial by multiplying the factors together. In this case, we can expand the polynomial as follows:

P(x) = (x - 3)x(x + 1)

P(x) = x(x^2 - 3x + x - 3)

P(x) = x(x^2 - 2x - 3)

P(x) = x^3 - 2x^2 - 3x

Q: How can we evaluate the polynomial at a given point?

A: We can evaluate the polynomial at a given point by substituting the value of $x$ into the polynomial. In this case, we can evaluate the polynomial at $x = 1$ as follows:

P(1) = (1)^3 - 2(1)^2 - 3(1)

P(1) = 1 - 2 - 3

P(1) = -4

Q: Which of the given functions could represent this graph?

A: The expanded polynomial matches function A: $x^3 - 2x^2 - 3x$ .

Q: What is the final answer?

A: The final answer is $\boxed{A}$ .

Additional Tips and Tricks

When dealing with polynomials, it's essential to understand the properties of the graph, including the $x$ -intercepts and the behavior of the polynomial.
The factors of a polynomial can be used to write the polynomial as a product of its factors.
Expanding a polynomial can help determine its behavior and properties.
Evaluating a polynomial at a given point can help determine if the polynomial passes through that point.

Final Answer

The final answer is: $\boxed{A}$