To solve the problem of finding the zeros of the polynomial function $f(x) = x^4 - 2x^3 + x^2 + 12x + 8$, you can use several methods as suggested: the Rational Zero Theorem, Descartes' Rule of Signs, and synthetic division or factoring.

Step 1: Rational Zero Theorem

This theorem suggests that potential rational roots can be found by taking the factors of the constant term (here $8$) and dividing them by the factors of the leading coefficient (here $1$). So the possible rational zeros are: $\pm 1, \pm 2, \pm 4, \pm 8$

Step 2: Descartes' Rule of Signs

This rule helps estimate the number of positive and negative real zeros by examining sign changes in the polynomial.

• For $f(x) = x^4 - 2x^3 + x^2 + 12x + 8$, the sign changes are:
  • From $x^4$ to $-2x^3$ (1 change),
  • From $-2x^3$ to $x^2$ (1 change),
  • From $x^2$ to $+12x$ (no change),
  • From $+12x$ to $+8$ (no change).

Thus, there are 2 sign changes, meaning there can be 2 or 0 positive real roots.

• For $f(-x) = (-x)^4 - 2(-x)^3 + (-x)^2 + 12(-x) + 8 = x^4 + 2x^3 + x^2 - 12x + 8$, the sign changes are:
  • From $x^4$ to $+2x^3$ (no change),
  • From $+2x^3$ to $x^2$ (no change),
  • From $x^2$ to $-12x$ (1 change),
  • From $-12x$ to $+8$ (1 change).

Thus, there are 2 sign changes, meaning there can be 2 or 0 negative real roots.

Step 3: Testing with Synthetic Division

Now, let's try one of the possible rational roots, say $x = 1$, using synthetic division.

1. Perform synthetic division of $f(x)$ by $x - 1$.
2. If the remainder is 0, $x = 1$ is a root, and you can factor $f(x)$ further.

Alternatively, using graphing tools can also help in locating an approximate root, after which further division or factoring can be used.

Would you like a detailed step-by-step synthetic division process for any potential root or further help on this? Let me know!

Related Questions:

1. How does Descartes' Rule of Signs work in determining possible zeros?
2. What is the Rational Zero Theorem, and how is it applied in this case?
3. What are the steps to perform synthetic division on polynomials?
4. How do you factor higher-degree polynomials like the one in this example?
5. How can graphing utilities help in identifying polynomial roots?

Tip:

Always check your potential rational zeros by substituting them into the polynomial or using synthetic division to verify whether they are actual roots.