Question

which polynomial expression in factored form has 3 as a leading coefficient with 0, 1, -3, and 8 as zeros?
a 3(x - 1)(x + 3)(x - 8)
b 3(x + 1)(x - 3)(x + 8)
c 3x(x - 1)(x + 3)(x - 8)
d 3x(x + 1)(x - 3)(x + 8)

Explanation:

Step1: Recall Factor Theorem

If \( r \) is a zero of a polynomial, then \( (x - r) \) is a factor. For zero \( 0 \), the factor is \( x - 0 = x \). For zero \( 1 \), the factor is \( (x - 1) \). For zero \( -3 \), the factor is \( (x - (-3))=(x + 3) \). For zero \( 8 \), the factor is \( (x - 8) \).

Step2: Include Leading Coefficient

The leading coefficient is \( 3 \), so we multiply the product of the factors by \( 3 \). Combining these, the polynomial in factored form is \( 3x(x - 1)(x + 3)(x - 8) \).

Step3: Match with Options

Compare with the given options. Option C is \( 3x(x - 1)(x + 3)(x - 8) \), which matches our derived polynomial.

Answer:

C. \( 3x(x - 1)(x + 3)(x - 8) \)