Questions: Given that 3 is a zero of the polynomial function (f(x)), find the remaining zeros. (f(x)=x^3-7 x^2+20 x-24) List the remaining zeros (other than 3). (Simplify your answer. Type an exact answer, using radicals and (i) as needed. Use a comma to separate answers.)

Given that 3 is a zero of the polynomial function (f(x)), find the remaining zeros.

(f(x)=x^3-7 x^2+20 x-24)

List the remaining zeros (other than 3).

(Simplify your answer. Type an exact answer, using radicals and (i) as needed. Use a comma to separate answers.)

Transcript text: Given that 3 is a zero of the polynomial function $f(x)$, find the remaining zeros. \[ f(x)=x^{3}-7 x^{2}+20 x-24 \] List the remaining zeros (other than 3 ). $\square$ (Simplify your answer. Type an exact answer, using radicals and $i$ as needed. Use a comma t

Solution

Solution Steps

To find the remaining zeros of the polynomial $ f(x) = x^3 - 7x^2 + 20x - 24 $, given that 3 is a zero, we can perform polynomial division to divide $ f(x) $ by $ (x - 3) $. This will give us a quadratic polynomial. We can then find the zeros of this quadratic polynomial using the quadratic formula.

Step 1: Polynomial Division

Given the polynomial $ f(x) = x^3 - 7x^2 + 20x - 24 $ and knowing that $ x = 3 $ is a zero, we perform polynomial division of $ f(x) $ by $ (x - 3) $. This results in a quadratic polynomial.

Step 2: Finding the Remaining Zeros

The quotient from the division is a quadratic polynomial, which we denote as $ g(x) $. We can find the zeros of $ g(x) $ using the quadratic formula. The remaining zeros are found to be:

\[ x = 2 - 2i \quad \text{and} \quad x = 2 + 2i \]