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)?
x=7
x equals 7
x=−5
x equals negative 5
x=−7
x equals negative 7
x=−53
The zero or root of a polynomial is the value of x that makes the polynomial equal to zero.
Using the Factor Theorem, if x-a is a factor of a polynomial function, then f(a) = 0.
In this case, g(x) = (x-7)(3x-5). To find a zero of g(x), we need to find a value of x that makes g(x) equal to zero.
Setting g(x) equal to zero, we have:
(x-7)(3x-5) = 0
To find the zeros, we set each factor equal to zero:
x-7 = 0 --> x = 7
3x-5 = 0 --> 3x = 5 --> x = 5/3
Therefore, x = 7 and x = 5/3 are zeros of g(x).
Out of the given options, x=7 is the zero of g(x).
To apply the Factor Theorem, we need to check which value of x makes g(x) equal to zero.
The polynomial g(x) is given as g(x) = (x - 7)(3x - 5).
To find the zero of g(x), we set g(x) equal to zero:
0 = (x - 7)(3x - 5).
Setting each factor equal to zero and solving for x, we have:
x - 7 = 0 or 3x - 5 = 0.
Solving these equations, we get:
x = 7 or x = 5/3.
Therefore, x = 7 is a zero of g(x).
To determine if a value is a zero of the polynomial g(x), we need to check if substituting that value into the polynomial results in zero.
Let's apply the Factor Theorem to each option:
1. For x = 7:
Substituting x = 7 into g(x) = (x - 7)(3x - 5):
g(7) = (7 - 7)(3(7) - 5)
g(7) = (0)(21 - 5)
g(7) = 0 * 16
g(7) = 0
Since g(7) = 0, x = 7 is a zero of g(x).
2. For x = -5:
Substituting x = -5 into g(x) = (x - 7)(3x - 5):
g(-5) = (-5 - 7)(3(-5) - 5)
g(-5) = (-12)(-15 - 5)
g(-5) = (-12)(-20)
g(-5) = 240
Since g(-5) is not equal to zero, x = -5 is not a zero of g(x).
3. For x = -7:
Substituting x = -7 into g(x) = (x - 7)(3x - 5):
g(-7) = (-7 - 7)(3(-7) - 5)
g(-7) = (-14)(-21 - 5)
g(-7) = (-14)(-26)
g(-7) = 364
Since g(-7) is not equal to zero, x = -7 is not a zero of g(x).
4. For x = -53:
Substituting x = -53 into g(x) = (x - 7)(3x - 5):
g(-53) = (-53 - 7)(3(-53) - 5)
g(-53) = (-60)(-159 - 5)
g(-53) = (-60)(-164)
g(-53) = 9840
Since g(-53) is not equal to zero, x = -53 is not a zero of g(x).
Therefore, the only zero of g(x) is x = 7.