Technique explanation
Argumental division applies the "first by first, last by last" principle to polynomial division. For exact division of a cubic by a linear factor, we can instantly determine the first and last terms of the quadratic quotient. The middle term is then found by comparing the coefficients of the dividend with the partial product of the known terms, bypassing the need for long division.
Standard fallback: Polynomial long division or synthetic division, where each power of x is reduced sequentially.
Worked examples
Example 1: Divide 6x³ + 11x² - 4x - 4 by 2x + 1
1. First Term: 6x³ / 2x = 3x².
2. Last Term: -4 / 1 = -4.
3. Middle Term: In (2x + 1)(3x² + nx - 4), the x² term is 2nx² + 3x². Set 2n + 3 = 11, so n = 4.
Result: 3x² + 4x - 4
Check: Multiply (2x + 1)(3x² + 4x - 4) = 2x(3x² + 4x - 4) + 1(3x² + 4x - 4) = (6x³ + 8x² - 8x) + (3x² + 4x - 4) = 6x³ + 11x² - 4x - 4. Match confirmed.
Example 2: Divide x³ + 6x² + 11x + 6 by x + 1
1. First Term: x³ / x = x².
2. Last Term: 6 / 1 = 6.
3. Middle Term: In (x + 1)(x² + nx + 6), the x² term is nx² + x². Set n + 1 = 6, so n = 5.
Result: x² + 5x + 6
Check: Multiply (x + 1)(x² + 5x + 6) = x(x² + 5x + 6) + 1(x² + 5x + 6) = (x³ + 5x² + 6x) + (x² + 5x + 6) = x³ + 6x² + 11x + 6. Match confirmed.
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