Coefficients from factor and remainder conditions

A factor condition gives an equation P(r)=0. A remainder condition gives an equation P(s)=R. Unknown coefficients can then be found by solving the resulting simultaneous equations. Why: polynomial division is an identity. At the zero of each linear divisor its quotient term vanishes, leaving just the stated remainder. Method: write the correct substitution for each condition; simplify independently; solve; substitute the values back into every original condition. Example: P(x)=x2+ax+b, with factor x-1 and remainder 6 on division by x+1. Then 1+a+b=0, 1-a+b=6. Subtraction gives a=-3, then b=2. Check: P(1)=0, P(-1)=6. Check: two copies of the same condition do not determine two unknowns. For divisor 2x-1, substitute x= 12.