What is the remainder when the polynomial 3x^4 - x^3 + 2x^2 - 4 is divided by x - 2?

To find the remainder when a polynomial is divided by a linear term such as ( x - 2 ), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial ( f(x) ) by ( x - c ) is simply ( f(c) ).

In this case, the given polynomial is ( f(x) = 3x^4 - x^3 + 2x^2 - 4 ) and we need to evaluate it at ( c = 2 ).

Calculating ( f(2) ):

1. Substitute ( x = 2 ) into the polynomial:

[

f(2) = 3(2^4) - (2^3) + 2(2^2) - 4

]

2. Calculate each term:

• ( 2^4 = 16 ) so ( 3(2^4) = 3 \times 16 = 48 )

• ( 2^3 = 8 ) so (-(2^3) = -8 )

• ( 2^2 = 4 ) so