Question

one zero of the polynomial function f(x) = x³ - 9x² + 20x is x = 0. what are the zeros of the polynomial function? 0, -5, -4 0, -5, 4 0, 5, -4 0, 5, 4

Explanation:

Step1: Factor the polynomial

Given \( f(x) = x^3 - 9x^2 + 20x \), we can factor out an \( x \) first: \( f(x)=x(x^2 - 9x + 20) \).

Step2: Factor the quadratic

Now, factor the quadratic \( x^2 - 9x + 20 \). We need two numbers that multiply to \( 20 \) and add to \( -9 \). Those numbers are \( -5 \) and \( -4 \)? Wait, no, wait: actually, \( x^2 - 9x + 20=(x - 5)(x - 4) \), because \( 5\times4 = 20 \) and \( 5 + 4 = 9 \). So the factored form is \( f(x)=x(x - 5)(x - 4) \).

Step3: Find the zeros

To find the zeros, set \( f(x)=0 \). So \( x = 0 \), \( x - 5 = 0 \) (which gives \( x = 5 \)), and \( x - 4 = 0 \) (which gives \( x = 4 \)). So the zeros are \( 0 \), \( 5 \), and \( 4 \).

Answer:

D. 0, 5, 4 (assuming the last option is D with 0, 5, 4; if the options are labeled as the last one being 0,5,4, then the answer is the option with 0, 5, 4, e.g., if the last option is "0, 5, 4", then the answer is that option.)