What is the remainder when 2x^4 + 19x^3 - 8x^2 - 11x + 4 is divided by x - 2?

To find the remainder when dividing the polynomial (2x^4 + 19x^3 - 8x^2 - 11x + 4) by (x - 2), you can apply the Remainder Theorem. The theorem states that the remainder of a polynomial (f(x)) when divided by (x - c) is given by (f(c)). In this case, (c = 2).

First, substitute (x = 2) into the polynomial:

[

f(2) = 2(2^4) + 19(2^3) - 8(2^2) - 11(2) + 4

]

Calculating each term:

• (2^4 = 16) and (2 \times 16 = 32)

• (2^3 = 8) and (19 \times 8 = 152)

• (2^2 = 4) and (-8 \times 4 = -32)

• (-11 \times 2 = -22)

Now, add these values together:

[

f(2) =