To determine the other zeros of the polynomial function [tex]\( f(x) = x^3 - 13x^2 + 60x - 100 \)[/tex], given that one of the zeros is [tex]\( x = 5 \)[/tex], follow these steps:
1. Identify the Polynomial and Known Zero:
- The polynomial is [tex]\( f(x) = x^3 - 13x^2 + 60x - 100 \)[/tex].
- The known zero of the polynomial is [tex]\( x = 5 \)[/tex].
2. Perform Polynomial Division:
- Since [tex]\( x = 5 \)[/tex] is a known zero, we can divide the polynomial [tex]\( f(x) \)[/tex] by [tex]\( x - 5 \)[/tex] to find the quotient polynomial.
- The division of [tex]\( x^3 - 13x^2 + 60x - 100 \)[/tex] by [tex]\( x - 5 \)[/tex] yields a quotient of [tex]\( x^2 - 8x + 20 \)[/tex].
3. Solve the Quotient Polynomial:
- Set the quotient polynomial equal to zero: [tex]\( x^2 - 8x + 20 = 0 \)[/tex].
- Solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 20 \)[/tex].
4. Calculate the Discriminant:
- The discriminant [tex]\( \Delta \)[/tex] is calculated as follows:
[tex]\( \Delta = b^2 - 4ac \)[/tex].
- Substituting the values, we get:
[tex]\[
\Delta = (-8)^2 - 4 \cdot 1 \cdot 20 = 64 - 80 = -16
\][/tex]
- Since the discriminant is negative ([tex]\( \Delta = -16 \)[/tex]), the solutions will be complex numbers.
5. Solve for the Complex Zeros:
- Using the quadratic formula with [tex]\( \Delta = -16 \)[/tex]:
[tex]\[
x = \frac{8 \pm \sqrt{-16}}{2 \cdot 1} = \frac{8 \pm 4i}{2} = 4 \pm 2i
\][/tex]
6. Identify the Other Zeros:
- The complex zeros are [tex]\( 4 - 2i \)[/tex] and [tex]\( 4 + 2i \)[/tex].
Therefore, the other zeros of the polynomial [tex]\( f(x) = x^3 - 13x^2 + 60x - 100 \)[/tex] are:
[tex]\[
4 - 2i, 4 + 2i
\][/tex]
