 2.Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases :
i) $$p(x) = 2x^3 + x^2 - 2x - 1$$, $$g(x) = x + 1$$
ii)$$p(x) = x^3 + 3x^2 + 3x + 1$$, $$g(x) = x + 2$$
iii)$$p(x) = x^3 - 4x^2 + x + 6$$, $$g(x) = x - 3$$

i)The zero of g(x) = x + 1 is –1.
Let $$p(x) = 2x^3 + x^2 - 2x - 1$$
So, To check, whether x + 1 is a factor of $$2x^3 + x^2 - 2x - 1$$.
By factor theorem,
$$p(–1) = 2{–1}^3 + {–1}^2 - 2{-1} - 1$$
i.e. $$p(–1) = –2 + 1 + 2 - 1$$
i.e.$$p(-1) = 0$$
Hence, g(x) = x + 1 is the factor of $$2x^3 + x^2 - 2x - 1$$

ii)The zero of $$g(x) = x + 2$$ is –2.
Let $$p(x) = x^3 + 3x^2 + 3x + 1$$
So, To check, whether x + 2 is a factor of $$x^3 + 3x^2 + 3x + 1$$.
By factor theorem,
$$p(–2) = {–2}^3 + 3{–2}^2 + 3{–2} + 1$$
i.e. $$p(–2) = –8 + 12 - 6 + 1$$
i.e.$$p(-2) = -1$$
Hence, $$g(x) = x + 2$$ is not the factor of $$x^3 + 3x^2 + 3x + 1$$

iii)The zero of $$g(x) = x - 3$$ is 3.
Let $$p(x) = x^3 - 4x^2 + x + 6$$
So, To check, whether x - 3 is a factor of $$p(x) = x^3 - 4x^2 + x + 6$$.
By factor theorem,
$$p(3) = {3}^3 - 4{3}^2 + {3} + 6$$
i.e. $$p(3) = 27 - 36 + 3 + 6$$
i.e.$$p(3) = 0$$
Hence, $$g(x) = x - 3$$ is the factor of $$p(x) = x^3 - 4x^2 + x + 6$$