Question

the roots of a polynomial equation are 0, 2, and -5. which of the following is a factored form of the equation with the given roots?

Explanation:

Step1: Recall Factor Theorem

If \( r \) is a root of a polynomial, then \( (x - r) \) is a factor.

Step2: Identify Factors from Roots

Given roots are \( 0 \), \( 2 \), and \( -5 \).

• For root \( 0 \): Factor is \( x - 0 = x \).
• For root \( 2 \): Factor is \( x - 2 \).
• For root \( -5 \): Factor is \( x - (-5) = x + 5 \).

Step3: Form Factored Polynomial

Multiply the factors: \( f(x)=x(x - 2)(x + 5) \).

Answer:

The factored form of the polynomial with roots \( 0 \), \( 2 \), and \( -5 \) is \( \boldsymbol{x(x - 2)(x + 5)} \) (or equivalent expanded/factored forms depending on options, but this is the general factored form from the roots).