Example:

Simplify the improper algebraic fraction:

Step 1: Polynomial Division

• Divide $\ ( x^3 ) \ by \ ( x )\ to\ get \ ( x^2 ).$
• Multiply$\ ( x^2 )\ by \ ( x + 1 )\ to\ get\ ( x^3 + x^2 )$.
• Subtract $\ x^3 + x^2$ from $\ x^3 + 2x^2 + 3x + 4$, which gives $\ x^2 + 3x + 4 .$
• Next, divide $\ x^2$ by $\ x$ to get $\ x .$
• Multiply $\ x$ by $\ x + 1$ to get $\ x^2 + x$.
• Subtract $\ x^2 + x$ from $\ x^2 + 3x + 4$, which gives $\ 2x + 4 .$
• Finally, divide $\ 2x$ by $\ x$ to get $\ 2$.
• Multiply $\ 2$ by $\ x + 1$ to get $\ 2x + 2 .$
• Subtract $\ 2x + 2$ from $\ 2x + 4$, leaving a remainder of $\ 2$. Result: $\frac{x^3 + 2x^2 + 3x + 4}{x + 1} = x^2 + x + 2 + \frac{2}{x + 1}$

So, the simplified form of the improper fraction is:

$x^2 + x + 2 + \frac{2}{x + 1}$

Example Problem:

Simplify the improper algebraic fraction:

Step-by-Step Solution:

Step 1: Perform polynomial division

To simplify the fraction $\frac{x^2 + 3x + 5}{x + 1}$, we need to divide the numerator $x^2 + 3x + 5$ by the denominator $x + 1$ using long division.

1. Divide the leading term of the numerator by the leading term of the denominator:

So, the first term of the quotient is $x$

2. Multiply $x$ by the denominator $x + 1$:

3. Subtract this from the original numerator:

4. Divide the leading term of the new expression by the leading term of the denominator:

So, the next term of the quotient is $2$.

5. Multiply $2$ by the denominator $x + 1$:

6. Subtract this from the remaining expression:

Step 2: Write the result as a mixed expression

At this point, we've completed the division, and the remainder is 3. So, we can write the improper fraction as:

Final Answer:

The simplified form of the improper algebraic fraction is:

Summary:

When simplifying an improper algebraic fraction, you divide the numerator by the denominator, and the result is a polynomial (the quotient) plus a proper fraction (the remainder divided by the denominator).