https://file.aiquickdraw.com/mathbot/file/1727054197328n47spohs.jpg

To address the problem provided in the image, we need to simplify the two rational functions of polynomials given:

### Part (a):
\[
H(s) = \frac{s^2 + 5s + 4}{s^3 + 13s^2 + 47s + 35}
\]
This expression represents a ratio of a quadratic polynomial to a cubic polynomial.

### Part (b):
\[
H(s) = \frac{s^5 + 5s^4 + 5s^3 + 5s^2 + 4s}{s^6 + 6s^5 + 12s^4 + 12s^3 + 11s^2 + 6s}
\]
This one is a ratio of a fifth-degree polynomial to a sixth-degree polynomial.

### Plan:
1. **Factorization**: Start by factoring both the numerator and denominator of each function.
2. **Cancellation**: After factoring, check if any terms cancel out.

Let's begin simplifying these expressions. I'll factorize the polynomials first.

### Part (a) Simplification
1. **Numerator of (a)**: \( s^2 + 5s + 4 \)
   - Factorize this quadratic polynomial.

2. **Denominator of (a)**: \( s^3 + 13s^2 + 47s + 35 \)
   - We will attempt to factor this cubic polynomial.

Let me proceed with these calculations.For **Part (a)**, after factorization, we have:

\[
\text{Numerator: } s^2 + 5s + 4 = (s + 1)(s + 4)
\]
\[
\text{Denominator: } s^3 + 13s^2 + 47s + 35 = (s + 1)(s + 5)(s + 7)
\]

Now, we can cancel the common factor of \( (s + 1) \) from both the numerator and denominator:

\[
H(s) = \frac{(s + 1)(s + 4)}{(s + 1)(s + 5)(s + 7)} = \frac{s + 4}{(s + 5)(s + 7)}
\]

Thus, the simplified form for **Part (a)** is:

\[
H(s) = \frac{s + 4}{(s + 5)(s + 7)}
\]

Now, let's move on to **Part (b)**. I'll factorize the polynomials for that part.For **Part (b)**, after factorization, we have:

\[
\text{Numerator: } s^5 + 5s^4 + 5s^3 + 5s^2 + 4s = s(s + 1)(s + 4)(s^2 + 1)
\]
\[
\text{Denominator: } s^6 + 6s^5 + 12s^4 + 12s^3 + 11s^2 + 6s = s(s + 1)(s + 2)(s + 3)(s^2 + 1)
\]

Now, we can cancel the common factors \( s \), \( (s + 1) \), and \( (s^2 + 1) \) from both the numerator and denominator:

\[
H(s) = \frac{s(s + 1)(s + 4)(s^2 + 1)}{s(s + 1)(s + 2)(s + 3)(s^2 + 1)} = \frac{s + 4}{(s + 2)(s + 3)}
\]

Thus, the simplified form for **Part (b)** is:

\[
H(s) = \frac{s + 4}{(s + 2)(s + 3)}
\]

### Final Simplified Forms:
1. **Part (a)**: \[
H(s) = \frac{s + 4}{(s + 5)(s + 7)}
\]
2. **Part (b)**: \[
H(s) = \frac{s + 4}{(s + 2)(s + 3)}
\]

Would you like further details on any of these steps?

### Related Questions:
1. What are the general methods for factorizing polynomials?
2. How can you check if a rational function has poles or zeroes on the complex plane?
3. What happens if the common factors in a ratio involve complex numbers?
4. How do you solve a system of equations using polynomial factorization?
5. What techniques can simplify higher-degree polynomials?

### Tip:
Always check for common factors before attempting complex polynomial division—it simplifies the problem significantly.

You have come across the following two ratios of two polynomials, but you’ve been told that they can be simplified. Simplify this ratio of two polynomials, but remember that the variables are complex, so be careful! You've been told that there are roots on the unit circle of complex numbers.

Learn how to simplify ratios of polynomials with complex roots, including step-by-step factorization and cancellation techniques. Understand polynomial behavior on the unit circle in complex numbers.

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