Write all real and imaginary zeros.
f(x)=x^3-8x^2+x-8
-8--1,-2,-4,-8
1
-+1,-+2,-+4,-+8
1 -8 1 -8
1
1 -7 -6
1 -7-6 -14
2 1 -8 1 -8
2 -12 -22
1 -6 -11 14
4 1 -8 1 -8
4 -16 -60
1 -4 -15 -52
8 1 -8 1 -8
8 0 8
1 0 1 0
answer: 8 is zero of f(x)
To find the real zeros and imaginary zeros of a polynomial function, we can use the Rational Root Theorem and synthetic division.
Let's apply the Rational Root Theorem first. The Rational Root Theorem states that if a polynomial function with integer coefficients has a rational root (𝑝/𝑞), where 𝑝 is a factor of the constant term and 𝑞 is a factor of the leading coefficient, then 𝑝 is a factor of the constant term and 𝑞 is a factor of the leading coefficient.
In our case, the polynomial function is f(x) = x^3 - 8x^2 + x - 8.
The constant term is -8, and the leading coefficient is 1. So, the possible rational roots are:
±1, ±2, ±4, ±8.
We can now proceed to check if any of these possible rational roots are actually roots of the polynomial by performing synthetic division.
Let's start with the possible root 1 using synthetic division:
1 -8 1 -8
-----------------
1 | 1 -7 -6 14
The result is not zero, so 1 is not a root.
Now, let's check -1:
-1 -8 1 -8
------------------
-1 | 1 -7 6 -14
Again, the result is not zero, so -1 is not a root.
Let's continue with the remaining possible rational roots.
For -2:
-2 -8 1 -8
-------------------
-2 | 1 -6 11 -14
The result is not zero, so -2 is not a root.
For 2:
2 -8 1 -8
-------------------
2 | 1 -6 -11 -14
The result is not zero, so 2 is not a root.
For -4:
-4 -8 1 -8
--------------------
-4 | 1 -4 -15 52
The result is not zero, so -4 is not a root.
For 4:
4 -8 1 -8
--------------------
4 | 1 0 -15 12
The result is not zero, so 4 is not a root.
For -8:
-8 -8 1 -8
--------------------
-8 | 1 8 0 64
The result is not zero, so -8 is not a root.
Finally, for 8:
8 -8 1 -8
--------------------
8 | 1 0 17 112
The result is zero, so 8 is a root.
Therefore, the real zero of the polynomial f(x) = x^3 - 8x^2 + x - 8 is 8.
Note that we checked all possible rational roots, and 8 is the only one that resulted in a zero remainder using synthetic division.