To find all the zeros of the polynomial [tex]\( f(x) = x^4 - 2x^3 - 7x^2 + 8x + 12 \)[/tex], we follow these steps:
1. Understand the Problem:
We need to find the values of [tex]\(x\)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. These are the roots or zeros of the polynomial.
2. Identify and Use Polynomial Theorems:
Polynomials of degree 4 can have up to 4 real roots. We will solve the polynomial equation [tex]\( x^4 - 2x^3 - 7x^2 + 8x + 12 = 0 \)[/tex].
3. Finding the Roots:
We solve the polynomial equation through algebraic means (e.g., factoring, synthetic division) or by using methods like the Rational Root Theorem to list possible rational roots and testing them. In this case, we determine the roots directly.
4. Verification:
Once we have potential roots, we verify them by substituting back into the polynomial to ensure they satisfy [tex]\( f(x) = 0 \)[/tex].
5. List the Roots:
After determining all the roots, we arrange them from smallest to largest, including any repeated roots as necessary.
By following these procedures, we find that the roots of the polynomial [tex]\( f(x) = x^4 - 2x^3 - 7x^2 + 8x + 12 \)[/tex] are:
[tex]\[
x = -2, -1, 2, 3
\][/tex]
These are all distinct roots, and there are no double roots to list twice. Therefore, arranging from smallest to largest, the roots are:
[tex]\[
x = [-2, -1, 2, 3]
\][/tex]
So, filling in the blanks:
[tex]\[
\begin{array}{c|c|c|c|c}
x & = & -2 & -1 & 2 & 3 \\
\end{array}
\][/tex]
