https://file.aiquickdraw.com/mathbot/file/1727497612952mry30m0x.jpg

The given polynomial is:

\[
P(x) = x^5 + 49x^3
\]

### Part (a): Find all zeros of \(P(x)\).

We can factor \(P(x)\) as follows:

\[
P(x) = x^3(x^2 + 49)
\]

Now, set \(P(x) = 0\):

\[
x^3(x^2 + 49) = 0
\]

This gives us two cases:

1. \(x^3 = 0 \Rightarrow x = 0\)
2. \(x^2 + 49 = 0 \Rightarrow x^2 = -49 \Rightarrow x = \pm 7i\)

Thus, the zeros of \(P(x)\) are:
\[
x = 0, 0, 0, 7i, -7i
\]

### Part (b): Factor \(P(x)\) completely.

From the factored form \(P(x) = x^3(x^2 + 49)\), we already have the complete factorization:

\[
P(x) = x^3(x - 7i)(x + 7i)
\]

This is the complete factorization of \(P(x)\).

Would you like more details or further explanation?

Here are 5 related questions you might find interesting:
1. How can the degree of a polynomial affect the number of complex roots?
2. What is the Fundamental Theorem of Algebra and how does it relate to this problem?
3. Can all polynomials be factored into real and complex factors?
4. How would the graph of this polynomial look based on its roots?
5. What are the implications of having multiple roots, like \(x = 0\) three times?

**Tip**: When factoring polynomials, always check for common factors first, as it simplifies the process significantly.

P(x) = x^5 + 49x^3. (a) Find all zeros of P, real and complex. (b) Factor P completely.

Learn how to find all real and complex zeros of the polynomial P(x) = x^5 + 49x^3, and factor it completely. This problem involves algebraic concepts like polynomial factoring and complex roots, making it suitable for students in Grades 10-12.

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