The function f, of, x, equals, 3, x, to the power 4 , minus, 26, x, cubed, plus, 68, x, squared, minus, 70, x, plus, 25f(x)=3x 
4
 −26x 
3
 +68x 
2
 −70x+25 has at least two rational roots. Use the rational root theorem to find those roots, then proceed to find all complex roots. (Note: roots may be integer, rational, irrational, and/or complex.)


To find the roots of the polynomial \( f(x) = 3x^4 - 26x^3 + 68x^2 - 70x + 25 \), we can apply the Rational Root Theorem, which states that any rational root, in the form \( \frac{p}{q} \), has \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient.

### Step 1: Identify Factors

- **Constant term**: \( 25 \)  
  Factors: \( \pm 1, \pm 5, \pm 25 \)

- **Leading coefficient**: \( 3 \)  
  Factors: \( \pm 1, \pm 3 \)

The possible rational roots (using \( \frac{p}{q} \)) are:
\[
\pm 1, \pm 5, \pm 25, \pm \frac{1}{3}, \pm \frac{5}{3}, \pm \frac{25}{3}
\]

### Step 2: Test Possible Rational Roots

We will test these possible roots in the polynomial \( f(x) \).

1. **Testing \( x = 1 \)**:
   \[
   f(1) = 3(1)^4 - 26(1)^3 + 68(1)^2 - 70(1) + 25 = 3 - 26 + 68 - 70 + 25 = 0
   \]
   \( x = 1 \) is a root.

2. **Testing \( x = 5 \)**:
   \[
   f(5) = 3(5)^4 - 26(5)^3 + 68(5)^2 - 70(5) + 25 
   \]
   Calculating this gives:
   \[
   = 3(625) - 26(125) + 68(25) - 350 + 25 = 1875 - 3250 + 1700 - 350 + 25 = -0
   \]
   \( x = 5 \) is also a root.

Now we have found two rational roots: \( x = 1 \) and \( x = 5 \).

### Step 3: Polynomial Division

Now we will perform synthetic division to reduce the polynomial using these roots.

1. **Divide \( f(x) \) by \( (x - 1) \)**:
   \[
   \begin{array}{r|rrrrr}
   1 & 3 & -26 & 68 & -70 & 25 \\
     &   & 3 & -23 & 45 & -25 \\
   \hline
     & 3 & -23 & 45 & -25 & 0 \\
   \end{array}
   \]
   Resulting polynomial: \( 3x^3 - 23x^2 + 45x - 25 \)

2. **Divide \( 3x^3 - 23x^2 + 45x - 25 \) by \( (x - 5) \)**:
   \[
   \begin{array}{r|rrrr}
   5 & 3 & -23 & 45 & -25 \\
     &   & 15 & -40 & 25 \\
   \hline
     & 3 & -8 & 5 & 0 \\
   \end{array}
   \]
   Resulting polynomial: \( 3x^2 - 8x + 5 \)

### Step 4: Solve the Quadratic

Now we solve \( 3x^2 - 8x + 5 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 3, b = -8, c = 5 \):
\[
x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot 5}}{2 \cdot 3} = \frac{8 \pm \sqrt{64 - 60}}{6} = \frac{8 \pm \sqrt{4}}{6} = \frac{8 \pm 2}{6}
\]
This gives us:
1. \( x = \frac{10}{6} = \frac{5}{3} \)
2. \( x = \frac{6}{6} = 1 \)

### Step 5: Summary of Roots

Thus, the roots of the polynomial are:
- Rational roots: \( x = 1, x = 5, x = \frac{5}{3} \)
- To find the complex roots, we already factored the polynomial down to a quadratic, which does not have additional irrational roots.

### Conclusion

The complete list of roots for \( f(x) = 3x^4 - 26x^3 + 68x^2 - 70x + 25 \):
- Rational roots: \( x = 1, x = 5, x = \frac{5}{3} \)

If you need further details or have questions about the process, let me know! Here are some follow-up questions to consider:

1. How can we graph this polynomial to visualize its roots?
2. What is the behavior of the polynomial as \( x \) approaches infinity?
3. How can we verify these roots using a different method?
4. What implications do the roots have on the function's local maxima and minima?
5. How can we use synthetic division to find other roots more efficiently? 

**Tip**: Always check for rational roots first before moving on to numerical methods for finding roots.

The function f(x) = 3x^4 - 26x^3 + 68x^2 - 70x + 25 has at least two rational roots. Use the rational root theorem to find those roots, then proceed to find all complex roots. (Note: roots may be integer, rational, irrational, and/or complex.)

Explore the step-by-step solution to find the roots of the polynomial f(x) = 3x^4 - 26x^3 + 68x^2 - 70x + 25 using the Rational Root Theorem. This guide covers identifying rational roots, performing synthetic division, and applying the quadratic formula. Suitable for high school students in Grades 9-12.

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