Testing factors and repeated factors

The factor theorem says ax+b is a factor of P(x) if and only if P(-b/a)=0, where a 0. Why: the remainder theorem gives remainder P(-b/a). A factor is precisely a divisor whose remainder is zero. Method: find the zero of the proposed factor, evaluate P, and state whether the remainder is zero. To test a repeated factor, divide once and test the quotient at the same zero. Example: P(x)=x2-x-2 has P(2)=0, so x-2 is a factor. Division gives P(x)=(x-2)(x+1). The quotient at 2 is 3, so (x-2)2 is not a factor. Check: testing P(2)=0 twice adds no evidence about multiplicity. The second test must use the quotient.