Given a polynomial: f(x) = x4 - 8x3 -8x2 +8x +7

Let's analyze and solve the polynomial \( f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7 \).

### Step-by-Step Solution

1. **Find the roots of the polynomial:**

   To find the roots of \( f(x) \), we need to solve the equation \( f(x) = 0 \), i.e.,

   \[
   x^4 - 8x^3 - 8x^2 + 8x + 7 = 0
   \]

2. **Check for Rational Roots:**

   According to the Rational Root Theorem, the possible rational roots of a polynomial \( ax^n + bx^{n-1} + \ldots + k = 0 \) are the factors of the constant term (\( k \)) divided by the factors of the leading coefficient (\( a \)). Here, the constant term is 7 and the leading coefficient is 1. The factors of 7 are \( \pm 1, \pm 7 \).

   So, the possible rational roots are \( \pm 1, \pm 7 \).

3. **Test Possible Rational Roots:**

   Let's test these possible roots by substituting them into the polynomial.

   - For \( x = 1 \):
     \[
     f(1) = 1^4 - 8(1)^3 - 8(1)^2 + 8(1) + 7 = 1 - 8 - 8 + 8 + 7 = 0
     \]
     Thus, \( x = 1 \) is a root.

4. **Factorize the Polynomial:**

   Since \( x = 1 \) is a root, we can factor \( (x - 1) \) out of \( f(x) \). Using synthetic division or polynomial long division, we divide \( x^4 - 8x^3 - 8x^2 + 8x + 7 \) by \( x - 1 \).

   - Synthetic Division:

     ```
     1 |  1   -8  -8   8   7
       |       1  -7 -15  -7
       ---------------------
         1   -7  -15  -7   0
     ```

     This gives us the quotient \( x^3 - 7x^2 - 15x - 7 \).

5. **Find the Roots of the Quotient Polynomial:**

   We now solve \( x^3 - 7x^2 - 15x - 7 = 0 \).

   - Check for rational roots using the Rational Root Theorem. The possible rational roots are \( \pm 1, \pm 7 \).

   - For \( x = 1 \):
     \[
     x^3 - 7x^2 - 15x - 7 = 1^3 - 7(1)^2 - 15(1) - 7 = 1 - 7 - 15 - 7 = -28 \neq 0
     \]

   - For \( x = -1 \):
     \[
     x^3 - 7x^2 - 15x - 7 = (-1)^3 - 7(-1)^2 - 15(-1) - 7 = -1 - 7 + 15 - 7 = 0
     \]
     Thus, \( x = -1 \) is a root.

6. **Further Factorization:**

   Since \( x = -1 \) is a root of \( x^3 - 7x^2 - 15x - 7 \), we can factor \( (x + 1) \) out of \( x^3 - 7x^2 - 15x - 7 \). Using synthetic division:

   ```
   -1 |  1  -7  -15  -7
      |     -1   8    7
      ----------------
         1  -8   -7   0
   ```

   This gives us the quotient \( x^2 - 8x - 7 \).

7. **Solve the Quadratic Equation:**

   Finally, we solve \( x^2 - 8x - 7 = 0 \).

   Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

   \[
   x = \frac{8 \pm \sqrt{64 + 28}}{2} = \frac{8 \pm \sqrt{92}}{2} = \frac{8 \pm 2\sqrt{23}}{2} = 4 \pm \sqrt{23}
   \]

### Summary of Roots

The roots of the polynomial \( f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7 \) are:
- \( x = 1 \)
- \( x = -1 \)
- \( x = 4 + \sqrt{23} \)
- \( x = 4 - \sqrt{23} \)

### Factorization of the Polynomial

The polynomial can be factorized as:
\[
f(x) = (x - 1)(x + 1)(x - (4 + \sqrt{23}))(x - (4 - \sqrt{23}))
\]

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you use synthetic division to factorize polynomials?
2. Can you explain the Rational Root Theorem in more detail?
3. How do you solve higher-degree polynomials?
4. What are some practical applications of finding polynomial roots?
5. How do you graph a polynomial function?

**Tip:** Always double-check your factorization by multiplying the factors to ensure you get back the original polynomial.

Learn how to solve the polynomial x^4 - 8x^3 - 8x^2 + 8x + 7 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomials, roots of polynomials, and the Rational Root Theorem. Suitable for advanced high school students (Grades 11-12).

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