Question

consider the polynomial function $f(x) = x^2 (x + 5)(x - 2)$. which of the following are zeros of the function? select all that apply. a -5 b -2 c 0 d 1 e 2 f 5

Explanation:

Step1: Recall zero - product property

The zero - product property states that if \(ab = 0\), then either \(a=0\) or \(b = 0\) (or both). For a polynomial function \(f(x)=x^{2}(x + 5)(x - 2)\), we set \(f(x)=0\), so \(x^{2}(x + 5)(x - 2)=0\).

Step2: Solve for \(x\) from each factor

• For the factor \(x^{2}=0\), taking the square root of both sides, we get \(x = 0\) (with multiplicity 2).
• For the factor \(x + 5=0\), subtracting 5 from both sides, we get \(x=-5\).
• For the factor \(x - 2=0\), adding 2 to both sides, we get \(x = 2\).

Answer:

A. \(-5\), C. \(0\), E. \(2\)