Solving quadratic equations by factoring: the AC metho dWe begin with an analysis of the quadratic equation whose leadingcoefficient is not unity and end with a few examples of the method.Suppose that a polynomial P (x)=Ax2+ Bx + C can be factoredinto two linear factors Mx + U and Nx + V , where the coefficients A,B, C,andM, U, N , V are all integers. ThenP (x)=(Mx + U)(Nx + V )=(Mx)(Nx + V )+U(Nx + V )= Mx Nx + MV x + UNx + UV= MNx2+ MV x + NUx + UV=(MN)x2+(MV + NU)x +(UV ) .Note thatA = MN ,B = MV + NU ,C = UV ,and thatA · C = MN · UV = MV · NU .This means that if we can find two integers m and n whose sum m+n =B and whose product mn = AC, then the original polynomial can befactored over the integers. Moreover, although not obvious, it is alsotrue that if no such pair of integers can be found, then the polynomialhas no factorization over the integers.1Solving quadratic equations by factoring: the AC metho dFor our first example, consider the polynomial x2+1. ThenA =1,B =0,andC =1. ThenA·C =1·1 = 1. But 1 = (1)·(1) = (−1)·(−1)are the only factorizations. Therefore x2+ 1 cannot be factored.For our second example, suppose that A = 1. Then the polynomialis x2+ Bx + C and The AC method tells us to look for a pair ofnumbers whose product is the constant term C and whose sum is thecoefficient of the middle term B. But this is the ordinary tecnique of”Trial and Error” you’ve always been using. So, the AC method isactually a generalization of what you have already been doing.For our third example, consider the polynomial 10z2− 13z − 3=0from Exercise 26 in Section 1.2 of Cohen, 6th edition. Then A =10and C = −3implythatA · C = −30. The question is, can you findtwo integers whose product is −30 and whose sum is −3? What about−30 = 2 · (−15) and −13 = 2 + (−15)? Yes, we can factor:10z2− 13z − 3=10z2+(2+(−15))z − 3=10z2+2z +(−15)z − 3=10z2+(−15)z +2z − 3=(10z − 15)z +(2z − 3)=5z(2z − 3) + (2z − 3) 1=(5z +1)(2z − 3) .The resulting factorization appears, like magic, by grouping terms andusing the distributive property!W2L1Patrick Dale McCray, 4 September
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