To find the zeros of the polynomial [tex]\( t(x) = x^3 - 3x^2 + 4x + 8 \)[/tex], we look for the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( t(x) = 0 \)[/tex].
Here are the steps to solve this:
1. Set the polynomial to zero:
[tex]\[
x^3 - 3x^2 + 4x + 8 = 0
\][/tex]
2. Identify possible rational roots:
The Rational Root Theorem suggests that any rational root, expressed as [tex]\(\frac{p}{q}\)[/tex], is a factor of the constant term (8) divided by a factor of the leading coefficient (1). Therefore, possible rational roots to test would be [tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex].
3. Test the possible rational roots:
By substituting these values into the polynomial, we find that [tex]\( x = -1 \)[/tex] is a root since it satisfies the equation.
4. Factor the polynomial using the found root:
Given that [tex]\( x = -1 \)[/tex] is a root, we can use polynomial division or synthetic division to divide [tex]\( t(x) \)[/tex] by [tex]\( (x + 1) \)[/tex]:
[tex]\[
t(x) = (x + 1)(x^2 - 4x + 8)
\][/tex]
5. Solve the quadratic equation [tex]\( x^2 - 4x + 8 = 0 \)[/tex]:
Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[
a = 1, \ b = -4, \ c = 8
\][/tex]
Substitute these values into the quadratic formula:
[tex]\[
x = \frac{4 \pm \sqrt{16 - 32}}{2} = \frac{4 \pm \sqrt{-16}}{2} = \frac{4 \pm 4i}{2} = 2 \pm 2i
\][/tex]
Therefore, the zeros of the polynomial [tex]\( t(x) = x^3 - 3x^2 + 4x + 8 \)[/tex] are:
[tex]\[
-1, \ 2 - 2i, \ 2 + 2i
\][/tex]
So, the zeros in exact form are [tex]\(-1, 2 - 2i, 2 + 2i\)[/tex].
