FactorizationDate: 03/17/99 at 22:34:02 From: karen Subject: Factorisation Can you help me simplify 9b^2 - 24bc + 16c^2? Date: 03/18/99 at 10:33:19 From: Doctor Rick Subject: Re: Factorisation Let us look at your goal. You want to get it in this form: 2 2 9b - 24bc + 16c = (Mb + Nc)(Pb + Qc) You need to replace the 4 letters M, N, P, and Q with numbers so you will get the right coefficients. The product MP must be 9, so M and P could be 1 and 9 or 3 and 3. (Order does not matter, since it is just a matter of which set of parentheses comes first.) The product NQ must be 16, so N and Q could be 1 and 16, 2 and 8, or 4 and 4. If M and P are 1 and 9 then order matters, so you would need to try 16 and 1, as well as 8 and 2. The sum of the inside and outside products (MQ + NP) will be -24 (remember the sign). Since it is negative, we must have at least one negative number, N or Q. But since NQ is positive, BOTH N and Q must be negative. There are quite a few possibilities to try. But since both 9 and 16 are perfect squares, my instinct says to try the squares first: M = 3 P = 3 N = 4 Q = 4 Now make that N = -4, Q = -4 since they must be negative. Try this out: (3b - 4c)(3b - 4c) = 9b^2 - 12bc ... 3b(3b-4c) - 12bc + 16c^2 ... -4c(2b-4c) ------------------- 9b^2 - 24bc + 16c^2 Sure enough, it works! Just for practice, let us see what would have happened if we tried something else first. What if M = 1, P = 9, and N = -8, Q = -2? MQ + NP = (1)(-2) + (-8)(9) = -2 - 72 = -74 That is not close to -24, so we would have to try another combination. As you do enough of these, you will develop a feel for what might work and what will not. For instance, you will get a smaller (absolute value) bc term if you put the bigger numbers either both first or both last, so they do not get multiplied together in the cross term (bc). With M = 1, P = 9, and N = -2, Q = -8: MQ + NP = (1)(-8) + (-2)(9) = -8 - 18 = -26 That is a lot closer to what we wanted (-24). My reasoning turned out to be correct. Of course, we already have the answer, and it is not very close to what I just tried. There is a lot of trial and error in factoring, but as I said, you can develop a feel for it. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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