*y*+1*y*−1​−(*y*+1)2(*y*−1)2​+*y*3+3*y*2+*y*+3​


To simplify the expression given by:

\[
\frac{y + 1}{y - 1} - \frac{(y + 1)^2}{(y - 1)^2} + y^3 + 3y^2 + y + 3,
\]

we'll approach it step by step.

### Step 1: Simplify the fractions

Let's start by simplifying the first two fractions:

1. **First Fraction:** \(\frac{y + 1}{y - 1}\) 

   This fraction is already in its simplest form.

2. **Second Fraction:** \(-\frac{(y + 1)^2}{(y - 1)^2}\)

   This fraction is also in its simplest form, though it can be noted that it represents a square of the first fraction.

### Step 2: Combine the fractions

Now, we need to combine the fractions over a common denominator:

\[
\frac{y + 1}{y - 1} - \frac{(y + 1)^2}{(y - 1)^2}
\]

To combine these, we rewrite the first fraction with the same denominator:

\[
\frac{(y + 1)(y - 1)}{(y - 1)(y - 1)} - \frac{(y + 1)^2}{(y - 1)^2}
\]

Simplify the numerator:

\[
\frac{y^2 - 1 - (y + 1)^2}{(y - 1)^2}
\]

Expand \((y + 1)^2\):

\[
(y + 1)^2 = y^2 + 2y + 1
\]

So, the expression becomes:

\[
\frac{y^2 - 1 - (y^2 + 2y + 1)}{(y - 1)^2}
\]

Simplify the numerator:

\[
\frac{y^2 - 1 - y^2 - 2y - 1}{(y - 1)^2} = \frac{-2y - 2}{(y - 1)^2}
\]

Factor out the \(-2\) in the numerator:

\[
\frac{-2(y + 1)}{(y - 1)^2}
\]

### Step 3: Combine with the remaining terms

Now, add the polynomial \(y^3 + 3y^2 + y + 3\) to this result:

\[
\frac{-2(y + 1)}{(y - 1)^2} + y^3 + 3y^2 + y + 3
\]

### Step 4: Interpret the expression

We will rewrite the entire expression:

\[
y^3 + 3y^2 + y + 3 + \frac{-2(y + 1)}{(y - 1)^2}
\]

This expression includes a polynomial part and a rational function part. 

### Conclusion

The expression simplifies to:

\[
y^3 + 3y^2 + y + 3 + \frac{-2(y + 1)}{(y - 1)^2}
\]

If you want to find critical points or values where the expression simplifies further, let me know!

Would you like more details, or have any questions?

### Follow-up Questions:

1. How can you find common denominators for combining fractions?
2. What techniques can be used to factor polynomials?
3. How do you determine the domain of a rational function?
4. How can polynomials be divided by other polynomials?
5. What are the steps to simplify complex rational expressions?

**Tip:** Always factor expressions completely and look for common denominators when combining fractions!

Learn how to simplify the rational expression (y + 1)/(y - 1) - ((y + 1)^2)/((y - 1)^2) + y^3 + 3y^2 + y + 3 step-by-step. This problem covers advanced algebraic concepts, including rational functions and polynomial simplification. Suitable for advanced high school students and beyond.

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