Question

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=−53
x equals negative Start Fraction 5 over 3 End Fraction

x=7
x equals 7

x=−7
x equals negative 7

x=−5
x equals negative 5

Answer 1

To determine which of the given values is a zero of the polynomial \( g(x) = (x - 7)(3x - 5) \), we can apply the Factor Theorem. According to the Factor Theorem, if \( x = a \) is a zero of the polynomial \( g(x) \), then \( g(a) = 0 \).

Let's check each option:

1. **\( x = -\frac{5}{3} \)**:
\[
g\left(-\frac{5}{3}\right) = \left(-\frac{5}{3} - 7\right)\left(3\left(-\frac{5}{3}\right) - 5\right)
\]
\[
= \left(-\frac{5}{3} - \frac{21}{3}\right)\left(-5 - 5\right) = \left(-\frac{26}{3}\right)(-10) \neq 0
\]

2. **\( x = 7 \)**:
\[
g(7) = (7 - 7)(3(7) - 5) = 0(21 - 5) = 0
\]
Thus, \( x = 7 \) is a zero.

3. **\( x = -7 \)**:
\[
g(-7) = (-7 - 7)(3(-7) - 5) = (-14)(-21 - 5) \neq 0
\]

4. **\( x = -5 \)**:
\[
g(-5) = (-5 - 7)(3(-5) - 5) = (-12)(-15 - 5) \neq 0
\]

After checking all the values, the only zero of \( g(x) \) is:

**\( x = 7 \)**.