Questions: Use the given zero to find the remaining zeros of the function. f(x)=x^3-4x^2+25x-100; zero: -5i

Transcript text: Use the given zero to find the remaining zeros of the function. \[ f(x)=x^{3}-4 x^{2}+25 x-100 ; \text { zero: }-5 i \]

Solution

Solution Steps

Step 1: Identify the Given Zero

The given zero of the polynomial \( f(x) = x^3 - 4x^2 + 25x - 100 \) is \( -5i \). Since complex zeros occur in conjugate pairs, the conjugate zero is \( 5i \).

Step 2: Form the Quadratic Factor

Using the given zero and its conjugate, we can form the quadratic factor: \[ (x - (-5i))(x - 5i) = (x + 5i)(x - 5i) = x^2 + 25 \]

Step 3: Divide the Polynomial

Next, we divide the original polynomial \( f(x) \) by the quadratic factor \( x^2 + 25 \): \[ f(x) = (x^2 + 25)(x - 4) \] This division yields the remaining factor \( x - 4 \).

Step 4: Solve for the Remaining Zero

Setting the remaining factor equal to zero gives us: \[ x - 4 = 0 \implies x = 4 \]