Answer: #3 x^4 - x^3 - 3 x^2 + 1# #=# #(x - 1)(3 x^3 + 2 x^2 - x - 1)#
Explanation:
Let's denote our polynomial #3 x^4 - x^3 - 3 x^2 + 1# by #P(x)#.
We use the Remainder Theorem: when dividing a polynomial #P(x)# by #(x-c)#, where #c# is a constant, the remainder equals #P(c)#.
Hence, for #x - c# to be a divisor of the polynomial #P(x)#, the remainder #P(c)# needs to be equal to zero.
Therefore, we need to find a constant #c# such that #P(c) = 0#.
#P(c) = 0# #<=># #3 c^4 - c^3 - 3 c^2 + 1 ##=# #0# #<=># #3 c^2 (c^2 - 1) - (c^3 - 1)##=##0# #<=># #3 c^2(c+1)(c-1) - (c - 1)(c^2 + c + 1)##=##0# #<=># #(c - 1)(3 c^3 + 3 c^2 - c^2 - c - 1)##=##0# #<=># #(c - 1)(3 c^3 + 2 c^2 - c - 1)##=##0#
So, we found that, if #c##=##1#, then #P(c) = 0#. Therefore, #x -1# is a divisor of the polynomial #P(x)#.
Verification : divide #P(x)##=##3 x^4 - x^3 - 3 x^2 + 1# by #x - 1# and obtain
#P(x)##/##(x - 1)##=##3 x^3 + 2 x^2 - x - 1#
