#Introduction to Polynomial Long Division

Polynomial long division is a method used to simplify rational functions, making them easier to integrate. For example, consider the rational function:

$$\frac{x^3 + 6x^2 - 9x - 11}{x-2}$$

Using polynomial long division, we can rewrite this function in a form that is easier to integrate.

#Steps for Polynomial Long Division

#Step 1: Setup

Identify the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by).

For example:

• Dividend: $x^3 + 6x^2 - 9x - 11$
• Divisor: $x - 2$

#Step 2: Divide the Leading Terms

Compare the highest power term in the dividend with the highest power term in the divisor.

$$x^3 \div x = x^2$$

Place $x^2$ on top of the division line.

#Step 3: Multiply and Subtract

Multiply the divisor by the term you just found and subtract it from the dividend.

$$x^2 \cdot (x - 2) = x^3 - 2x^2$$ Subtract from the dividend:

$$(x^3 + 6x^2 - 9x - 11) - (x^3 - 2x^2) = 8x^2 - 9x - 11$$

#Step 4: Repeat

Repeat the process with the new polynomial.

Divide: $$8x^2 \div x = 8x$$

Multiply and Subtract: $$8x \cdot (x - 2) = 8x^2 - 16x$$ Subtract:

$$(8x^2 - 9x - 11) - (8x^2 - 16x) = 7x - 11$$

Divide: $$7x \div x = 7$$

Multiply and Subtract: $$7 \cdot (x - 2) = 7x - 14$$ Subtract:

$$(7x - 11) - (7x - 14) = 3$$

The remainder is 3. ### Final Answer Combine all the terms including the remainder:

$$\frac{x^3 + 6x^2 - 9x - 11}{x-2} = x^2 + 8x + 7 + \frac{3}{x-2}$$

#Worked Example

#Problem

Find the indefinite integral:

$$\int \frac{x^3 + 2x + 13}{x + 3} , dx$$

#Solution

Start by using polynomial long division. Add $0x^2$ as a placeholder for the missing $x^2$ term:

$$\frac{x^3 + 2x + 13}{x + 3} = x^2 - 3x + 11 - \frac{20}{x + 3}$$

Now integrate term by term:

$$\int \left( x^2 - 3x + 11 - \frac{20}{x + 3} \right) , dx$$

$$= \int x^2 , dx - \int 3x , dx + \int 11 , dx - \int \frac{20}{x + 3} , dx$$

$$= \frac{1}{3} x^3 - \frac{3}{2} x^2 + 11x - 20 \ln|x + 3| + C$$

#Answer

$$\int \frac{x^3 + 2x + 13}{x + 3} , dx = \frac{1}{3} x^3 - \frac{3}{2} x^2 + 11x - 20 \ln|x + 3| + C$$

#Glossary

• Dividend: The polynomial being divided.
• Divisor: The polynomial by which the dividend is divided.
• Remainder: The leftover part of the dividend after division.
• Indefinite Integral: The general form of the antiderivative, including a constant of integration $C$.

#Summary and Key Takeaways

#Summary

Polynomial long division simplifies rational functions, making them easier to integrate. The process involves dividing, multiplying, and subtracting terms in a step-by-step manner.

#Key Takeaways

• Use polynomial long division to simplify rational functions.
• Always compare the highest power terms during division.
• Be meticulous with signs and subtraction to avoid errors.
• Practice using polynomial long division to become proficient.