Polynomial Long Division with nonlinear divisor
Related Readings (Optional): long division for decimals in the site section on Number Theory.
Exercise: For the above long division, check that
| divisor | d(x) = x2-3x+3 |
| dividend | p(x) = 3x5+4x4-5x3-3x2 +3x+6 |
| quotient | q(x) = 3x3+13x2+26x+42 |
| remainder | r(x) = 45x -120 |
implies p(x) = d(x)q(x) +r(x). If the latter equality holds, the long division above is correct - free of errors.
A Polynomial Division Theorem. If p(x) is a polynomial of degree n > 0 and d(x) is nonzero polynomial of degree m > 0 , then there exists a polynomial q(x) of degree max(0, n-r) and real number r(x) of degree m-1 or less such that p(x) = q(x) d(x) + r(x). Moreover r(x) = 0 (the zero polynomial) when and only when p(x) is a polynomial multiple of d(x).
This theorem can be proven by the method of mathematical induction, a proof method met in or before calculus. Since that method is not at our disposable, we will give a few examples to indicate how the polynomial quotient polynomial q(x) and the remainder r (a real number or a polynomial of degree 0) can be calculated. That is sufficient in practice.
Exercise: Use the polynomial long division to find polynomials a(x) (the quotient) and r(x) (the remainder) such that
x4+3x3+6x2 + 4x + 2= a(x)(x2 + 2x + 1) + r(x)
with degree (r(x)) < 2. In other words, apply the long division algorithm (method) to dividend x4+3x3+6x2 + 4x + 2 using x2 + 2x + 1 as a divisor.