AlgMap-13.1 [[input:ans]] [[validation:ans]]]]> As can be seen by multiplying out a general, factorised quadratic: \[ (cx+a)(dx+b) = cdx^2+(da+cb)x+ab, \] \(a\) and \(b\) are factors of the constant term, and \(c\) and \(d\) are factors of the \(x^2\) coefficient so there are only a finite number of (integer) cases that are possible. With practice it should be simple enough to find \(a\), \(b\), \(c\) and \(d\). In this example: \[ {@expand((cf1*v+a)*(cf2*v+b))@}. \] Equating with the general quadratic above gives: \[{cd = {@cf1*cf2@}}.\] \[{da+cb = {@cf2*a+cf1*b@}}.\] \[{ab = {@a*b@}}.\] The correct values are: \(a = {@a@}\), and \(b = {@b@}\), \(c = {@cf1@}\) and, \(d = {@cf2@}\). By substituting these back into the general quadratic, and factorising constants out, we obtain the factored form: \[{@ta@}.\] 1 0.1 0 v:rand([x,y,z,t]); cf1:rand(4)+1; cf2:(rand(4)+1)*rand([1,-1]); a:(rand(9)+1)*rand([1,-1]); b:(rand(9)+1)*rand([1,-1]); ta:factor((cf1*v+a)*(cf2*v+b)); [[feedback:prt1]]

]]> \[ {@expand((cf1*v+a)*(cf2*v+b))@} = {@ta@} \] 1 0 0 *10 none 1 i cos-1 lang [ 0 {"author":[{"firstName":"Tony","lastName":"Barnard","institution":"","year":"1999"}],"language":["en"],"isPartOf":"STACK Library - Algebra Refresher","license":"cc-sa-4.0","additional":{"UoE":{"Notes":"This is automatically generated metadata."}},"freeform":{}} ans algebraic ta 15 1 0 0 1 1 1 1 1 prt1 1.0000000 1 1 0 FacForm ans ta v 0 = 1.0000000 -1 1-0-T = 0.0000000 -1 1-0-F 1874421384 1770560704 1402158234 1601177468 1592030075 786618329 1948945848 1985214913 607236540 222685436 1 ans ta prt1 1.0000000 0.0000000 1-0-T 2 ans ev(expand((cf1*v+a)*(cf2*v+b)),simp) prt1 0.0000000 0.1000000 1-0-F