+288 Let f(x) be a polynomial such that f(0)=4, f(1)=5, and f(2)=10. Find the remainder when f(x) is divided by x(x-1)(x-2).
----Thanks!
+33614 The remainder will, in general, be a polynomial of order less than that of the divisor (otherwise it would still be divisible by the divisor). So
remainder = u*x2 + v*x + w
The non-remainder part vanishes when x=0, 1 or 2, so
f(0) = 4 = u*0 +v*0 + w, or w = 4
f(1) = 5 = u*1 + v*1 + 4, or u + v = 1
f(2) = 10 = u*4 + v*2 + 4 or 4u + 2v = 6
From these last two equations we get u = 2 and v = -1
Remainder = 2x2 - x + 4
+33614 The remainder will, in general, be a polynomial of order less than that of the divisor (otherwise it would still be divisible by the divisor). So
remainder = u*x2 + v*x + w
The non-remainder part vanishes when x=0, 1 or 2, so
f(0) = 4 = u*0 +v*0 + w, or w = 4
f(1) = 5 = u*1 + v*1 + 4, or u + v = 1
f(2) = 10 = u*4 + v*2 + 4 or 4u + 2v = 6
From these last two equations we get u = 2 and v = -1
Remainder = 2x2 - x + 4
+118608 Thanks Alan,
I am still not understanding this statement
"The non-remainder part vanishes when x=0, 1 or 2, "
Could you expand upon that please?
