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!

madyl Mar 11, 2020

#2**+4 **

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*x^{2} + 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 = 2x^{2} - x + 4

Alan Mar 12, 2020

#2**+4 **

Best Answer

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*x^{2} + 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 = 2x^{2} - x + 4

Alan Mar 12, 2020

#4**0 **

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?

Melody
Mar 13, 2020