Question

select the correct answer. if the zeros of a quadratic function are 3 and 8, what are the factors of the function? a. (x + 8) and (x - 3) b. (x - 8) and (x + 3) c. (x + 8) and (x + 3) d. (x - 8) and (x - 3)

Explanation:

Step1: Recall the factor theorem

The factor theorem states that if \( r \) is a zero of a polynomial function \( f(x) \), then \( (x - r) \) is a factor of \( f(x) \).

Step2: Find the factors for each zero

For the zero \( 3 \), using the factor theorem, the corresponding factor is \( (x - 3) \)? Wait, no. Wait, if \( r = 3 \), then the factor is \( (x - r)=(x - 3) \)? Wait, no, wait. Wait, let's correct. If the zero is \( r \), then \( f(r)=0 \), so \( (x - r) \) is a factor. So for zero \( 3 \), the factor is \( (x - 3) \)? Wait, no, wait. Wait, let's take an example. Suppose the zero is \( 2 \), then the factor is \( (x - 2) \) because when \( x = 2 \), \( (x - 2)=0 \). So for zero \( 3 \), the factor is \( (x - 3) \)? Wait, no, wait the options. Wait, the zeros are \( 3 \) and \( 8 \). So for zero \( 3 \), the factor is \( (x - 3) \)? Wait, no, wait the options. Wait, let's check the options. Wait, option D is \( (x - 8) \) and \( (x - 3) \). Let's see: if \( x = 8 \), then \( (x - 8)=0 \), and if \( x = 3 \), then \( (x - 3)=0 \). So according to the factor theorem, if \( 8 \) is a zero, then \( (x - 8) \) is a factor, and if \( 3 \) is a zero, then \( (x - 3) \) is a factor. So the factors are \( (x - 8) \) and \( (x - 3) \), which is option D.

Answer:

D. \((x - 8)\) and \((x - 3)\)