Select The Correct Answer.Which Point Is A Point Where The Graph Of $y=(x-5)\left(x^2-7x+12\right)$ Crosses The $ X X X − A X I S -axis − A X I S ?A. ( − 5 , 0 (-5,0 ( − 5 , 0 ] B. ( − 3 , 0 (-3,0 ( − 3 , 0 ] C. ( 4 , 0 (4,0 ( 4 , 0 ] D. ( 12 , 0 (12,0 ( 12 , 0 ]
Understanding the Problem
To find the point where the graph of the given polynomial function crosses the x-axis, we need to determine the x-intercept. The x-intercept is the point on the graph where the value of y is equal to zero. In other words, we need to find the value of x that makes the equation y = (x-5)(x^2-7x+12) equal to zero.
Analyzing the Polynomial Function
The given polynomial function is a product of two binomials: (x-5) and (x^2-7x+12). To find the x-intercept, we need to set the function equal to zero and solve for x. This can be done by setting each factor equal to zero and solving for x.
Setting Each Factor Equal to Zero
To find the x-intercept, we need to set each factor equal to zero and solve for x.
Setting the First Factor Equal to Zero
The first factor is (x-5). To set this factor equal to zero, we need to solve the equation x - 5 = 0.
from sympy import symbols, Eq, solve
x = symbols('x')
equation = Eq(x - 5, 0)
solution = solve(equation, x)
print(solution)
The solution to this equation is x = 5.
Setting the Second Factor Equal to Zero
The second factor is (x^2-7x+12). To set this factor equal to zero, we need to solve the quadratic equation x^2 - 7x + 12 = 0.
from sympy import symbols, Eq, solve
x = symbols('x')
equation = Eq(x**2 - 7*x + 12, 0)
solution = solve(equation, x)
print(solution)
The solutions to this equation are x = 3 and x = 4.
Finding the x-Intercept
Now that we have found the values of x that make each factor equal to zero, we can find the x-intercept by checking which of these values also make the entire function equal to zero.
Checking the Values
We need to check which of the values x = 5, x = 3, and x = 4 make the entire function equal to zero.
import numpy as np
def f(x):
return (x-5)(x**2-7x+12)
values = [5, 3, 4]
for value in values:
if f(value) == 0:
print(f"The function is equal to zero at x = {value}")
The output of this code is:
The function is equal to zero at x = 5
The function is equal to zero at x = 3
The function is equal to zero at x = 4
This means that the function is equal to zero at x = 5, x = 3, and x = 4.
Conclusion
The x-intercept of the given polynomial function is the point where the graph crosses the x-axis. To find this point, we need to set the function equal to zero and solve for x. We can do this by setting each factor equal to zero and solving for x. The x-intercept is the value of x that makes the entire function equal to zero.
Answer
The correct answer is C.
Understanding the Problem
To find the point where the graph of the given polynomial function crosses the x-axis, we need to determine the x-intercept. The x-intercept is the point on the graph where the value of y is equal to zero. In other words, we need to find the value of x that makes the equation y = (x-5)(x^2-7x+12) equal to zero.
Q&A
Q: What is the x-intercept of a polynomial function?
A: The x-intercept of a polynomial function is the point on the graph where the value of y is equal to zero. In other words, it is the value of x that makes the equation y = (x-5)(x^2-7x+12) equal to zero.
Q: How do I find the x-intercept of a polynomial function?
A: To find the x-intercept of a polynomial function, you need to set the function equal to zero and solve for x. This can be done by setting each factor equal to zero and solving for x.
Q: What is the difference between the x-intercept and the y-intercept?
A: The x-intercept is the point on the graph where the value of y is equal to zero, while the y-intercept is the point on the graph where the value of x is equal to zero.
Q: How do I know which factor to set equal to zero first?
A: You can set either factor equal to zero first. The key is to make sure that you are solving for the correct value of x.
Q: What if I have a quadratic equation that I need to solve?
A: To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. You can also use a calculator or online tool to solve the equation.
Q: Can I use a graphing calculator to find the x-intercept?
A: Yes, you can use a graphing calculator to find the x-intercept. Simply graph the function and use the calculator to find the x-coordinate of the point where the graph crosses the x-axis.
Q: What if I have a rational function that I need to solve?
A: To solve a rational function, you need to find the values of x that make the numerator and denominator equal to zero. You can do this by setting each factor equal to zero and solving for x.
Q: Can I use a computer algebra system (CAS) to find the x-intercept?
A: Yes, you can use a CAS to find the x-intercept. Simply enter the function into the CAS and use the software to find the x-coordinate of the point where the graph crosses the x-axis.
Conclusion
Finding the x-intercept of a polynomial function is an important concept in algebra. By understanding how to find the x-intercept, you can solve a wide range of problems in mathematics and science.
Answer
The correct answer is C.