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

If \( r \) is a zero of a polynomial function, then \( (x - r) \) is a factor of the polynomial.

Step2: Apply the factor theorem to each zero

For the zero \( 3 \), the factor is \( (x - 3) \)? Wait, no. Wait, if the zero is \( 3 \), then \( x = 3 \) implies \( x - 3 = 0 \), so the factor is \( (x - 3) \)? Wait, no, wait. Wait, if the zero is \( 3 \), then \( x = 3 \) so \( x - 3 = 0 \), so the factor is \( (x - 3) \)? Wait, no, wait, the zero is \( 3 \), so the factor is \( (x - 3) \)? Wait, no, wait, let's check again. Wait, the zero is \( 3 \), so when \( x = 3 \), the factor should be zero. So \( x - 3 = 0 \) when \( x = 3 \), so the factor is \( (x - 3) \). Similarly, the zero is \( 8 \), so when \( x = 8 \), the factor should be zero, so \( x - 8 = 0 \) when \( x = 8 \), so the factor is \( (x - 8) \). Wait, but let's check the options. Option D is \( (x - 8) \) and \( (x - 3) \). Wait, let's verify each option:

• Option A: \( (x + 8) \) and \( (x - 3) \). If we set \( x + 8 = 0 \), \( x = -8 \); \( x - 3 = 0 \), \( x = 3 \). So zeros at -8 and 3. Not 3 and 8.
• Option B: \( (x - 8) \) and \( (x + 3) \). Zeros at 8 and -3. Not 3 and 8.
• Option C: \( (x + 8) \) and \( (x + 3) \). Zeros at -8 and -3. Not 3 and 8.
• Option D: \( (x - 8) \) and \( (x - 3) \). Zeros at 8 and 3. Which matches the given zeros.

Wait, so the correct factors are \( (x - 8) \) and \( (x - 3) \), which is option D.

Answer:

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