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=−5 x equals negative 5 x=7 x equals 7 x=−7 x equals negative 7 x=−53

Answer 1

Given a polynomial f(x) and that f(12)=0, apply the Remainder Theorem and then determine which of the following is a factor of f(x).(1 point) Responses (x−32) left parenthesis x minus Start Fraction 3 over 2 End Fraction right parenthesis (x−12) left parenthesis x minus 12 right parenthesis (x+12) left parenthesis x plus 12 right parenthesis (x−0) left parenthesis x minus 0 right parenthesis

Answer 2

To find the zeroes of the polynomial \( g(x) = (x - 7)(3x - 5) \), we can use the Factor Theorem, which states that if \( (x - c) \) is a factor of a polynomial, then \( c \) is a zero of that polynomial.

We can set each factor equal to zero to find the zeroes:

1. From the factor \( (x - 7) = 0 \):
\[
x - 7 = 0 \Rightarrow x = 7
\]

2. From the factor \( (3x - 5) = 0 \):
\[
3x - 5 = 0 \Rightarrow 3x = 5 \Rightarrow x = \frac{5}{3}
\]

Therefore, the zeroes of the polynomial \( g(x) \) are \( x = 7 \) and \( x = \frac{5}{3} \).

Among the given choices, the zero of \( g(x) \) is \( x = 7 \).

Answer 3

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( (x - c) \), then the remainder of that division is \( f(c) \). If \( f(c) = 0 \), then \( (x - c) \) is a factor of \( f(x) \).

Given that \( f(12) = 0 \), we can apply the Remainder Theorem:

Since \( f(12) = 0 \), it follows that \( (x - 12) \) is a factor of \( f(x) \).

Among the options provided, the correct factor of \( f(x) \) is:

- \( (x - 12) \)

So, the answer is \( (x - 12) \).