https://file.aiquickdraw.com/mathbot/file/1730157432120g8o434n1.jpg

Let's solve the given polynomial problem.

The polynomial is:
\[
P(x) = x^6 - 7x^3 - 8
\]

### Part (a): Finding All Zeros of \( P(x) \)
To find all zeros, let's make a substitution to simplify:
1. Let \( u = x^3 \), so \( u^2 = x^6 \).
2. Then \( P(x) \) becomes:
   \[
   P(u) = u^2 - 7u - 8
   \]
3. Now we solve \( u^2 - 7u - 8 = 0 \).

This is a quadratic equation, so we use the quadratic formula:
\[
u = \frac{7 \pm \sqrt{49 + 32}}{2} = \frac{7 \pm \sqrt{81}}{2} = \frac{7 \pm 9}{2}
\]
This gives us:
\[
u = 8 \quad \text{and} \quad u = -1
\]

4. Substitute back \( u = x^3 \):
   - For \( u = 8 \): \( x^3 = 8 \Rightarrow x = \sqrt[3]{8} = 2 \).
   - For \( u = -1 \): \( x^3 = -1 \Rightarrow x = \sqrt[3]{-1} = -1 \).

So, the real zeros of \( P(x) \) are \( x = 2 \) and \( x = -1 \).

5. Since \( x = 2 \) has a cube root, there’s only one value, and similarly for \( x = -1 \).

### Part (b): Factoring \( P(x) \) Completely
Now we can express \( P(x) \) as:
\[
P(x) = (x^3 - 2)(x^3 + 1)
\]
We can factor each term further:
- \( x^3 - 2 \) is already in simplest form (irreducible over real numbers).
- \( x^3 + 1 \) can be factored using the sum of cubes formula:
  \[
  x^3 + 1 = (x + 1)(x^2 - x + 1)
  \]

Thus, the complete factorization of \( P(x) \) is:
\[
P(x) = (x^3 - 2)(x + 1)(x^2 - x + 1)
\]

Would you like further details on any steps?

### Related Questions
1. How do you find complex roots of a polynomial?
2. What is the difference between real and complex factors?
3. How do you apply the quadratic formula in polynomial problems?
4. What other substitutions could simplify higher-degree polynomials?
5. How do you handle irreducible polynomials over real numbers?

**Tip:** When factoring polynomials, look for substitutions that can reduce the polynomial to a lower degree.

A polynomial P(x) = x^6 - 7x^3 - 8 is given. (a) Find all zeros of P, real and complex. (b) Factor P completely.

Solve the polynomial P(x) = x^6 - 7x^3 - 8 by finding all real and complex zeros, and factor it completely. Learn about polynomial factorization, substitution methods, and the quadratic and sum of cubes formulas. Suitable for Grades 10-12 students.

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