Polynomial Long Division: Dividing P(x) by D(x)

Dividing Polynomials Using Long Division

This example demonstrates how to divide polynomial P(x) by polynomial D(x) using long division:

Given:

• P(x) = 3x⁵ + x³ - 3x² + 4x - 9
• D(x) = x² - 4x + 1

Solution:

We set up the long division problem like we would with regular numbers:

                  3x³ + 12x² + 44x + 169
          ____________________________________
x² - 4x + 1 | 3x⁵ + 0x⁴ - 3x³ + 4x² + 0x - 9
                  - (3x⁵ - 12x⁴ + 3x³)
          ____________________________________
                       12x⁴ - 7x³ + 4x² + 0x - 9
                       - (12x⁴ - 48x³ + 12x²)
          ____________________________________
                                41x³ - 8x² + 0x - 9
                                - (41x³ - 164x² + 41x)
          ____________________________________
                                        156x² - 41x - 9

Result:

Therefore, the quotient P(x)/D(x) can be expressed as:

P(x)/D(x) = 3x³ + 12x² + 44x + 169 + (156x² - 41x - 9)/(x² - 4x + 1)

This means:

• Q(x) = 3x³ + 12x² + 44x + 169 is the quotient
• R(x) = 156x² - 41x - 9 is the remainder
• D(x) = x² - 4x + 1 is the divisor