To determine the points where the graph of the function [tex]\( y = (x-5)(x^2 - 7x + 12) \)[/tex] crosses the [tex]\( x \)[/tex]-axis, we need to find the values of [tex]\( x \)[/tex] that make [tex]\( y = 0 \)[/tex].
The function [tex]\( y \)[/tex] is given by:
[tex]\[ y = (x-5)(x^2 - 7x + 12) \][/tex]
To find where the function crosses the [tex]\( x \)[/tex]-axis, we need to set [tex]\( y \)[/tex] to 0 and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = (x-5)(x^2 - 7x + 12) \][/tex]
Next, we solve the equation by setting each factor equal to zero:
1. [tex]\( x - 5 = 0 \)[/tex]
2. [tex]\( x^2 - 7x + 12 = 0 \)[/tex]
Solving the first equation:
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
Solving the second equation [tex]\( x^2 - 7x + 12 = 0 \)[/tex]:
We can factorize the quadratic equation [tex]\( x^2 - 7x + 12 \)[/tex]:
[tex]\[ x^2 - 7x + 12 = (x - 3)(x - 4) \][/tex]
So, we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 3 = 0 \quad \text{or} \quad x - 4 = 0 \][/tex]
[tex]\[ x = 3 \quad \text{or} \quad x = 4 \][/tex]
Thus, the solutions to the equation [tex]\( (x-5)(x^2 - 7x + 12) = 0 \)[/tex] are:
[tex]\[ x = 3, \; x = 4, \; x = 5 \][/tex]
These values of [tex]\( x \)[/tex] give the points where the graph crosses the [tex]\( x \)[/tex]-axis. Thus, the points are [tex]\( (3, 0) \)[/tex], [tex]\( (4, 0) \)[/tex], and [tex]\( (5, 0) \)[/tex].
Given the options:
A. [tex]\((-5,0)\)[/tex]
B. [tex]\((-3,0)\)[/tex]
C. [tex]\((4,0)\)[/tex]
D. [tex]\((12,0)\)[/tex]
The correct answer is:
C. [tex]\( (4, 0) \)[/tex]
