Use the method of long division to explain if (x-1) is a factor of 3x^4-5x^3+12x^2-8x+9

Answer 1

a much easier method is just plug x=1 into the 4th degree polynomial, you

get: 3-5+12 -8 +9 = 11 which is not 0, so x-1 is not a factor

but long division leads to a remainder of 11, x-1 does not divide evenly. the quotient is 3x^3-2x^2+10x+2 with remainder of 11 or 11/(x-1)

You first divide x into 3x^4 to get 3x^3, then multiply 3x^3 by -1 to get -3x^3. subtract that from -5x^3 to get -2x^3. then divide x into -2x^3 to get -2x^2. multiply -2x^2 by -1 to get 2x^2. subtract 2x^2 from 12x^2 to get 10x^2. divide x into 10x^2 to get 10x. multiply 10x by -1 to get -10x. subtract -10x from -8x to get 2x. divide x into 2x to get 2. multiply 2 by-1 to get -2. subtract -2 from 9 to get the remainder of 11. Once you get a remainder you know x-1 is not a factor and 1 is not a zero.

I'm not sure, but my guess is this 4th degree polynomial has no factors or zeroes. Just plugging in a few simple integers seems to make the graph always above the x axis. a 4th degree polynomial with positive coefficient for the 4th degree term tends to look like a W and if the two local minimums are above zero, there are no factors.