Division of Unknown PolynomialsDate: 3/18/96 at 15:6:36 From: Anonymous Subject: Algebra problem I am an eighth grade student at the Willmar Junior High School in Willmar, Minnesota. I have a math question that I need help on. When a polynomial P(x) is divided by x-1, the remainder is 3. When P(x)is divided by x-2, the remainder is 5. Find the remainder when P(x) is divided by x^2-3x+2. Thank you, Jacob Date: 3/22/96 at 19:28:51 From: Doctor Steven Subject: Re: Algebra problem This is a tough problem for an eighth grade class! First let's look at what we know: 1. We know P(x) when divided by (x-1) gives a remainder of 3. So P(x) = f(x)*(x-1) + 3. (We don't care what f(x) is) 2. We know P(x) when divided by (x-2) gives a remainder of 5. So P(x) = g(x)*(x-2) + 5, also.(We don't care what g(x) is either!) So P(x) = f(x)*(x-1) + 3 = g(x)*(x-2) + 5. Subtract 5 from every part of this three part equation to get: P(x) - 5 = f(x)*(x-1) - 2 = g(x)*(x-2) + 0. Well, we also know that (x-1) = 1*(x-2) + 1. So the remainder of x-1 when divided by x-2 is +1. Now check this out for some higher mathematics (really though it's pretty easy): ____ ___ Look at 4| 5 its remainder is 1. Now look at 4| 3 its remainder is 3. ____ Now look at 4| 15 its remainder is 3, or 3*1 the product of the remainders of 5/4 and 3/4. This works for anything that is multiplied together. So f(x)*(x-1) - 2 better not have a remainder when divided by (x-2) (since it equals the far righthand side, which definitely doesn't have a remainder when divided by (x-2) ). This tells us that f(x) better have a remainder of 2 when divided by (x-2) (since rem(2)*rem(1) - rem(2) = rem(0) ). So f(x) = m(x)*(x-2) + 2. (We don't care what m(x) is) Now we have: P(x) - 5 = [m(x)*(x-2) + 2](x-1) - 2 = g(x)(x-2) + 0. But all we need to look at now is: P(x) - 5 = [m(x)*(x-2) + 2](x-1) - 2. Add 5 to both sides to get: P(x) = [m(x)*(x-2) + 2](x-1) + 3. Then simplify on the right to get: P(x) = m(x)*(x-2)*(x-1) + (2x - 2) + 3. Simplify some more to get: P(x) = m(x)*(x^2 - 3x + 2) + 2x + 1. So the remainder of P(x) when divided by x^2 - 3x + 2 is (2x + 1) Whew! That was a toughy. ;) -Doctor Steven, The Math Forum |
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