Given the polynomial g(x) in its factored form as shown here: g(x)=(x−7)(3x−5) , apply the Factor Theorem. Which of the following is a zero of g(x) ?(1 point)
Responses
x=7
x equals 7
x=−5
x equals negative 5
x=−7
x equals negative 7
x=−53
None of the given options are correct.
To apply the Factor Theorem, we need to find the value of x that makes the polynomial g(x) equal to zero.
Given that g(x) is in its factored form as (x−7)(3x−5), we can set it equal to zero:
(x−7)(3x−5) = 0
Now, to find the zeros of g(x), we solve this equation by setting each factor equal to zero:
x - 7 = 0 --> x = 7
and
3x - 5 = 0 --> 3x = 5 --> x = 5/3
Therefore, the zero of g(x) is x = 7.
To determine which of the given options is a zero of the polynomial g(x), we need to evaluate g(x) using each option and check if the result is zero.
The Factor Theorem states that if a polynomial g(x) can be factored as (x - a) times some other polynomial, then a is a zero of g(x).
Given that the polynomial g(x) is in factored form as g(x) = (x - 7)(3x - 5), we can substitute each value from the options into g(x) and check if the result is zero.
Substituting x = 7:
g(7) = (7 - 7)(3(7) - 5) = (0)(21 - 5) = 0.
So, x = 7 is a zero of g(x).
Substituting x = -5:
g(-5) = (-5 - 7)(3(-5) - 5) = (-12)(-15 - 5) = (-12)(-20) = 240.
So, x = -5 is not a zero of g(x).
Substituting x = -7:
g(-7) = (-7 - 7)(3(-7) - 5) = (-14)(-21 - 5) = (-14)(-26) = 364.
So, x = -7 is not a zero of g(x).
Substituting x = -53:
g(-53) = (-53 - 7)(3(-53) - 5) = (-60)(-159 - 5) = (-60)(-164) = 9840.
So, x = -53 is not a zero of g(x).
Therefore, the only zero of g(x) is x = 7.