Learning Guide Learning Guide

at_expr := (x^16 - y^16) / (x^8 - y^8);5.1 Mathematical Manipulations • 151rat_expr := x16 − y 16x 8 − y 8> factor( rat_expr );x 8 + y 8> rat_expr := (x^16 - y^16) / (x^7 - y^7);rat_expr := x16 − y 16x 7 − y 7> factor(rat_expr);(y + x) (x 2 + y 2 ) (x 4 + y 4 ) (x 8 + y 8 )x 6 + y x 5 + y 2 x 4 + y 3 x 3 + y 4 x 2 + y 5 x + y 6Specifying the Algebraic Number Field The factor command factors apolynomial over the ring implied by the coefficients. The following polynomialhas integer coefficients, so the terms in the factored form haveinteger coefficients.> poly := x^5 - x^4 - x^3 + x^2 - 2*x + 2;poly := x 5 − x 4 − x 3 + x 2 − 2 x + 2> factor( poly );(x − 1) (x 2 − 2) (x 2 + 1)In this next example, the coefficients include √ 2. Note the differencesin the result.> expand( sqrt(2)*poly );√2 x 5 − √ 2 x 4 − √ 2 x 3 + √ 2 x 2 − 2 √ 2 x + 2 √ 2> factor( % );