To convert the expression $\frac{y^2}{y+1}$ into the form $y - 1 + \frac{1}{y+1}$, let's break it down step-by-step:

Step 1: Perform Polynomial Division

We need to divide $y^2$ by $y + 1$. To do this, we use polynomial long division.

1. Divide the first term: Divide the leading term of $y^2$ by the leading term of $y+1$, which gives: $\frac{y^2}{y} = y$ So, the first term of the quotient is $y$.

2. Multiply and subtract: Now multiply $y$ by $y + 1$: $y \cdot (y + 1) = y^2 + y$ Subtract this from $y^2$: $y^2 - (y^2 + y) = -y$ So, we are left with $-y$.

3. Repeat the process: Now divide the new leading term $-y$ by $y$: $\frac{-y}{y} = -1$ So, the next term in the quotient is $-1$.

4. Multiply and subtract again: Multiply $-1$ by $y + 1$: $-1 \cdot (y + 1) = -y - 1$ Subtract this from $-y$: $-y - (-y - 1) = 1$ So, the remainder is $1$.

Step 2: Write the Final Result

Now we can express the original fraction as: $\frac{y^2}{y+1} = y - 1 + \frac{1}{y+1}$

This is the desired form. So, the division yields the expression $y - 1 + \frac{1}{y+1}$.

Recap of the Steps:

• Polynomial division of $\frac{y^2}{y+1}$ gives $y - 1 + \frac{1}{y+1}$.

Do you need any further details or have any questions on this process?

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Here are some related questions to help deepen your understanding:

1. What is the result when dividing a cubic expression by a linear factor?
2. How do you perform synthetic division for a polynomial?
3. What is the significance of the remainder in polynomial division?
4. How can we apply polynomial division to simplify rational expressions?
5. Can you divide a polynomial by a quadratic term? How would the process differ?

Tip: Always check your division by multiplying the quotient by the divisor to see if you recover the original expression.