Select The Correct Answer.Which Ordered Pair Represents A Point Where The Graph Of F(x)=(x-1)\left(x^2+x-20\right ] Crosses The X X X -axis?A. ( − 5 , 0 (-5,0 ( − 5 , 0 ]B. ( − 4 , 0 (-4,0 ( − 4 , 0 ]C. ( − 1 , 0 (-1,0 ( − 1 , 0 ]D. ( 20 , 0 (20,0 ( 20 , 0 ]

Understanding the Problem

To find the point where the graph of a function crosses the x-axis, we need to determine the value of x at which the function equals zero. In other words, we are looking for the x-intercept of the graph. The given function is $f(x)=(x-1)\left(x^2+x-20\right)$, and we want to find the ordered pair that represents a point where the graph of this function crosses the x-axis.

The x-Axis and Function Intercepts

The x-axis is a horizontal line that intersects the graph of a function at its x-intercepts. At these points, the value of the function is zero, and the graph touches the x-axis. To find the x-intercepts of a graph, we need to set the function equal to zero and solve for x.

Solving for x

To find the x-intercepts of the graph of $f(x)=(x-1)\left(x^2+x-20\right)$, we need to set the function equal to zero and solve for x. This can be done by setting each factor of the function equal to zero and solving for x.

Factoring the Quadratic Expression

The quadratic expression $x^2+x-20$ can be factored as $(x+5)(x-4)$. Therefore, the function can be written as $f(x)=(x-1)\left((x+5)(x-4)\right)$.

Setting Each Factor Equal to Zero

To find the x-intercepts of the graph, we need to set each factor of the function equal to zero and solve for x. Setting the first factor equal to zero gives us $x-1=0$, which implies that $x=1$. Setting the second factor equal to zero gives us $(x+5)(x-4)=0$, which implies that $x=-5$ or $x=4$.

Checking the Solutions

We need to check each of the solutions to make sure that they are valid. Plugging $x=1$ into the function gives us $f(1)=(1-1)\left((1+5)(1-4)\right)=0$, which is a valid solution. Plugging $x=-5$ into the function gives us $f(-5)=(-5-1)\left((-5+5)(-5-4)\right)=0$, which is also a valid solution. Plugging $x=4$ into the function gives us $f(4)=(4-1)\left((4+5)(4-4)\right)=0$, which is a valid solution.

The Correct Answer

Based on our calculations, we have found three valid solutions: $x=1$, $x=-5$, and $x=4$. However, we are only interested in the point where the graph crosses the x-axis, which is the point with the lowest x-coordinate. Therefore, the correct answer is $(-5,0)$.

Conclusion

In this article, we have discussed how to find the point where the graph of a function crosses the x-axis. We have used the given function $f(x)=(x-1)\left(x^2+x-20\right)$ and set it equal to zero to find the x-intercepts of the graph. We have also checked each of the solutions to make sure that they are valid. Based on our calculations, we have found that the correct answer is $(-5,0)$.

Final Answer

The final answer is $\boxed{(-5,0)}$.

Understanding the Problem

To find the point where the graph of a function crosses the x-axis, we need to determine the value of x at which the function equals zero. In other words, we are looking for the x-intercept of the graph. The given function is $f(x)=(x-1)\left(x^2+x-20\right)$, and we want to find the ordered pair that represents a point where the graph of this function crosses the x-axis.

Q&A Session

Q: What is the x-axis?

A: The x-axis is a horizontal line that intersects the graph of a function at its x-intercepts. At these points, the value of the function is zero, and the graph touches the x-axis.

Q: How do I find the x-intercepts of a graph?

A: To find the x-intercepts of a graph, we need to set the function equal to zero and solve for x. This can be done by setting each factor of the function equal to zero and solving for x.

Q: What is the difference between a factor and a term?

A: A factor is an expression that is multiplied together to form a product, while a term is a single expression that is added or subtracted from another expression.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, we need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What is the significance of the x-intercepts of a graph?

A: The x-intercepts of a graph represent the points where the graph crosses the x-axis. These points are important because they give us information about the behavior of the function.

Q: How do I check my solutions to make sure they are valid?

A: To check your solutions, you need to plug each solution into the function and make sure that the result is zero.

Q: What is the correct answer for the given problem?

A: Based on our calculations, we have found three valid solutions: $x=1$, $x=-5$, and $x=4$. However, we are only interested in the point where the graph crosses the x-axis, which is the point with the lowest x-coordinate. Therefore, the correct answer is $(-5,0)$.

Conclusion

In this article, we have discussed how to find the point where the graph of a function crosses the x-axis. We have used the given function $f(x)=(x-1)\left(x^2+x-20\right)$ and set it equal to zero to find the x-intercepts of the graph. We have also checked each of the solutions to make sure that they are valid. Based on our calculations, we have found that the correct answer is $(-5,0)$.

Final Answer

The final answer is $\boxed{(-5,0)}$.

Additional Resources

• For more information on finding the x-intercepts of a graph, please see our article on "Finding the x-Intercepts of a Graph".
• For more information on factoring quadratic expressions, please see our article on "Factoring Quadratic Expressions".
• For more information on checking solutions, please see our article on "Checking Solutions".