we know that
The zeros of the function are the values of x when the value of the function is equal to zero
so
Let's verify each case to determine the solution.
Substitute the value of [tex]x=3[/tex] and [tex]x=7[/tex] in each function and then verify the value of the function
case A) [tex]f(x)=x^{2} +4x-21[/tex]
For [tex]x=3[/tex]
[tex]f(3)=3^{2} +4*3-21=0[/tex]
therefore
the value of [tex]x=3[/tex] is a zero of the function
For [tex]x=7[/tex]
[tex]f(7)=7^{2} +4*7-21=56[/tex]
therefore
the value of [tex]x=7[/tex] is not a zero of the function
The case A) is not the solution
case B) [tex]f(x)=x^{2} -4x-21[/tex]
For [tex]x=3[/tex]
[tex]f(3)=3^{2} -4*3-21=-24[/tex]
therefore
the value of [tex]x=3[/tex] is not a zero of the function
For [tex]x=7[/tex]
[tex]f(7)=7^{2} -4*7-21=0[/tex]
therefore
the value of [tex]x=7[/tex] is a zero of the function
The case B) is not the solution
case C) [tex]f(x)=x^{2} -10x+21[/tex]
For [tex]x=3[/tex]
[tex]f(3)=3^{2} -10*3+21=0[/tex]
therefore
the value of [tex]x=3[/tex] is a zero of the function
For [tex]x=7[/tex]
[tex]f(7)=7^{2} -10*7+21=0[/tex]
therefore
the value of [tex]x=7[/tex] is a zero of the function
The case C) is a solution
case D) [tex]f(x)=x^{2} -10x-21[/tex]
For [tex]x=3[/tex]
[tex]f(3)=3^{2} -10*3-21=-42[/tex]
therefore
the value of [tex]x=3[/tex] is not a zero of the function
For [tex]x=7[/tex]
[tex]f(7)=7^{2} -10*7-21=-42[/tex]
therefore
the value of [tex]x=7[/tex] is not a zero of the function
The case D) is not a solution
therefore
The answer is
[tex]f(x)=x^{2} -10x+21[/tex]
