Find all zeroes of the function f(x)=4x^4-12x^3-33x^2+89x-84

To find all the zeroes of the function \( f(x) = 4x^4 - 12x^3 - 33x^2 + 89x - 84 \), we will use the following steps:

### 1. Rational Root Theorem
The Rational Root Theorem suggests possible rational roots in the form \( \frac{p}{q} \), where \( p \) are the factors of the constant term (-84) and \( q \) are the factors of the leading coefficient (4).

Factors of -84: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 7, \pm 12, \pm 14, \pm 21, \pm 28, \pm 42, \pm 84 \)  
Factors of 4: \( \pm 1, \pm 2, \pm 4 \)

Possible rational roots: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 4, \pm 6, \pm 7, \pm \frac{7}{2}, \pm 12, \pm 14, \pm 21, \pm 28, \pm 42, \pm 84 \)

### 2. Test Possible Rational Roots
We will test some of these rational roots by substituting them into the function \( f(x) \).

Let's begin with \( x = 2 \):
\[
f(2) = 4(2)^4 - 12(2)^3 - 33(2)^2 + 89(2) - 84 = 4(16) - 12(8) - 33(4) + 89(2) - 84
\]
\[
= 64 - 96 - 132 + 178 - 84 = 0
\]
So, \( x = 2 \) is a root.

### 3. Polynomial Division
Now that we know \( x = 2 \) is a root, we can divide \( f(x) \) by \( x - 2 \) to find the other factors.

Using synthetic division:

\[
\begin{array}{r|rrrrr}
2 & 4  & -12  & -33  & 89  & -84  \\
  &    & 8    & -8   & -82 & 14   \\
\hline
  & 4  & -4   & -41  & 7   & 0   \\
\end{array}
\]

The quotient is \( 4x^3 - 4x^2 - 41x + 7 \).

### 4. Solve the Quotient Polynomial
We now solve the cubic equation \( 4x^3 - 4x^2 - 41x + 7 = 0 \). We can try factoring or using numerical methods, such as the Rational Root Theorem again.

Testing \( x = -1 \):
\[
4(-1)^3 - 4(-1)^2 - 41(-1) + 7 = -4 - 4 + 41 + 7 = 40 \neq 0
\]
Testing \( x = \frac{1}{2} \) or others doesn't yield zeros either, so I will calculate the exact zeros numerically.

The roots of the cubic polynomial \( 4x^3 - 4x^2 - 41x + 7 = 0 \) are approximately:

\[
x_1 \approx 3.666, \quad x_2 \approx -2.834, \quad x_3 \approx 0.168
\]

### 5. Final Zeroes of the Function
The complete set of zeroes of the original function \( f(x) = 4x^4 - 12x^3 - 33x^2 + 89x - 84 \) is:

\[
x = 2, \quad x \approx 3.666, \quad x \approx -2.834, \quad x \approx 0.168
\]

Would you like more details on any step?

Here are some related questions to expand your understanding:

1. What is the Rational Root Theorem and how is it used?
2. How do you perform synthetic division with polynomials?
3. What are the methods to solve cubic polynomials analytically?
4. How can numerical methods find the roots of higher-degree polynomials?
5. What other techniques exist for finding polynomial roots aside from the Rational Root Theorem?

**Tip:** Polynomial division simplifies equations by reducing the degree, making them easier to solve.

Discover how to find all zeroes of the quartic polynomial function f(x) = 4x^4 - 12x^3 - 33x^2 + 89x - 84 using the Rational Root Theorem and synthetic division. This detailed solution includes testing possible rational roots and solving a cubic equation. Perfect for students in advanced algebra or pre-calculus.

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