Respuesta :
The function is
[tex]f(x)=x^4+6x^3-3x^2-52x-60[/tex]
The "zeros" of a function, are the values of x, for which f(x)=0
According to the "Rational Root Theorem", the rational roots of f, are factors of 60.
That is, when trying to find the roots of a polynomial function, it is a very good idea to first check the factors of the constant term.
All numbers shown in the choices are factors of 60, so we will solve the problem by trial:
[tex]f(2)=2^4+6\cdot2^3-3\cdot2^2-52\cdot2-60=16+48-12-104-60 \neq 0[/tex]
2 is not a root, so we eliminate choices A and D,
Choices B and C are almost equal, with only 1 different number, so let's check whether 5 is a root or not:
[tex]f(5)=(5)^4+6\cdot(5)^3-3\cdot(5)^2-52\cdot(5)-60[/tex]
[tex]=625+750-75-160-60=60[/tex]
but
[tex]f(-5)=(-5)^4+6\cdot(-5)^3-3\cdot(-5)^2-52\cdot(-5)-60[/tex]
[tex]=625-750-75+160-60=0[/tex]
So the right choice is B
[tex]f(x)=x^4+6x^3-3x^2-52x-60[/tex]
The "zeros" of a function, are the values of x, for which f(x)=0
According to the "Rational Root Theorem", the rational roots of f, are factors of 60.
That is, when trying to find the roots of a polynomial function, it is a very good idea to first check the factors of the constant term.
All numbers shown in the choices are factors of 60, so we will solve the problem by trial:
[tex]f(2)=2^4+6\cdot2^3-3\cdot2^2-52\cdot2-60=16+48-12-104-60 \neq 0[/tex]
2 is not a root, so we eliminate choices A and D,
Choices B and C are almost equal, with only 1 different number, so let's check whether 5 is a root or not:
[tex]f(5)=(5)^4+6\cdot(5)^3-3\cdot(5)^2-52\cdot(5)-60[/tex]
[tex]=625+750-75-160-60=60[/tex]
but
[tex]f(-5)=(-5)^4+6\cdot(-5)^3-3\cdot(-5)^2-52\cdot(-5)-60[/tex]
[tex]=625-750-75+160-60=0[/tex]
So the right choice is B
