Question

Suppose that f(x) = -x^3 - 3x^2 - 2x + 1 and g(x) = 2x^3 + x^2 + x + 1. Find f(x) + g(x). Hence, factorise f(x) + g(x) completely given that (x-1) is one of its factors.

A. f(x) + g(x) = x^3 - 2x - 1, Factorized form: (x - 1)(x^2 + x + 1)
B. f(x) + g(x) = -x^2 - 2x + 2, Factorized form: (x - 1)(x^2 + 2)
C. f(x) + g(x) = x^3 - 3x^2 - 2x + 1, Factorized form: (x - 1)(x^2 + x + 1)
D. f(x) + g(x) = -x^3 - 3x^2 - x + 2, Factorized form: (x - 1)(x^2 + x + 1)

Answer 1

Upon adding f(x) and g(x) we get f(x) + g(x) = x^3 - 2x^2 - x + 2. However, (x - 1) is not a factor of this polynomial as it does not satisfy the polynomial (remainder is nonzero). None of the provided options are correct, indicating an error in the given question.

Step-by-step explanation:

To find f(x) + g(x), we need to add the two given polynomials:

f(x) = -x^3 - 3x^2 - 2x + 1
g(x) = 2x^3 + x^2 + x + 1

Summing these, we get:

f(x) + g(x) = (-x^3 + 2x^3) + (-3x^2 + x^2) + (-2x + x) + (1 + 1)
f(x) + g(x) = x^3 - 2x^2 - x + 2

Given that (x - 1) is a factor of f(x) + g(x), we can use polynomial division to factorize the resultant polynomial. However, upon checking, we find that (x - 1) is not a factor of x^3 - 2x^2 - x + 2 as it does not satisfy the polynomial (the remainder is not zero).

None of the given options for factorization are correct because (x - 1) is not a factor of the polynomial we have found, meaning we cannot factorize it in the form presented in the options. It seems there might be an error in the given problem.