PolynomialReduce¶

Reduces a polynomial modulo a list of polynomials in a single variable. PolynomialReduce[poly, {p1, p2, ...}, x] returns {{a1, a2, ...}, b} with a1 p1 + a2 p2 + ... + b == poly and b the minimal remainder.

$ wo 'PolynomialReduce[x^2 + 1, {x + 1}, x]'
{{-1 + x}, 2}

$ wo 'PolynomialReduce[x^3 + 2 x + 1, {x^2 + 1, x + 1}, x]'
{{x, 1}, 0}

$ wo 'PolynomialReduce[x^3, {x - 1}, x]'
{{1 + x + x^2}, 1}

$ wo 'PolynomialReduce[x^2, {3 x + 1}, x]'
{{-1/9 + x/3}, 1/9}

A divisor with rational coefficients divides just as exactly. The factor that clears the denominator stays in front of the quotient rather than being distributed into it:

$ wo 'PolynomialReduce[1 - 2 x^3 + x^4, {(1 + x + x^2 - x^3)/2}, x]'
{{2*(1 - x)}, 0}

A single divisor may be given without the surrounding list:

$ wo 'PolynomialReduce[x^2 + 1, x + 1, x]'
{{-1 + x}, 2}

When no divisor can reduce the polynomial, it is returned as the remainder:

$ wo 'PolynomialReduce[x^2 + 1, {x^3 + 1}, x]'
{{0}, 1 + x^2}