Polynomial Division

Level 1 - Math I (Physics) topic page in Integration.

Polynomial Long Division

When integrating rational functions where the numerator's degree is greater than or equal to the denominator's degree, polynomial long division simplifies the integrand.

When to Use

For \(\int \frac{P(x)}{Q(x)} \, dx\) where \(\deg(P) \geq \deg(Q)\), divide first to get a polynomial plus a proper fraction.

Polynomial Long Division Steps

Arrange both polynomials in descending order
Divide the leading terms: (leading of numerator) ÷ (leading of denominator)
Multiply the divisor by this result, subtract from numerator
Repeat until degree of remainder < degree of divisor

Example

Divide \(2x^3 + 4x^2 - 3x + 1\) by \(x^2 + 1\):

Division Setup

\[\begin{aligned} 2x^3 + 4x^2 - 3x + 1 &\div (x^2 + 1) \end{aligned}\]

Step 1: \(2x^3 \div x^2 = 2x\)

Multiply: \(2x(x^2 + 1) = 2x^3 + 2x\)

Subtract: \((2x^3 + 4x^2 - 3x + 1) - (2x^3 + 2x) = 4x^2 - 5x + 1\)

Step 2: \(4x^2 \div x^2 = 4\)

Multiply: \(4(x^2 + 1) = 4x^2 + 4\)

Subtract: \((4x^2 - 5x + 1) - (4x^2 + 4) = -5x - 3\)

Result

\[2x^3 + 4x^2 - 3x + 1 = (x^2 + 1)(2x + 4) + (-5x - 3)\]

So: \(\frac{2x^3 + 4x^2 - 3x + 1}{x^2 + 1} = 2x + 4 + \frac{-5x - 3}{x^2 + 1}\)

Integration Application

Integration

\[\int \frac{2x^3 + 4x^2 - 3x + 1}{x^2 + 1} \, dx = \int (2x + 4) \, dx + \int \frac{-5x - 3}{x^2 + 1} \, dx\]

Result

\[= x^2 + 4x - \frac{5}{2}\ln(x^2+1) - 3\arctan x + C\]

This technique makes otherwise difficult rational integrals manageable.

Substitution

Partial Fractions