Several academic document-sharing sites host PDFs specifically for the Barnard and Child exercises:
The authors originally published a separate book titled which acts as the official key. higher algebra by barnard and child solutions pdf
Do not search for the full PDF. Instead, search for “Barnard and Child, problem X.X” on Math Stack Exchange. Many problems from Chapter V (Theory of Equations) or Chapter X (Determinants) have been discussed in detail. You get rigorous explanations, often better than a bare solution. search for “Barnard and Child