 1.Determine which of the following polynomials has a factor :
i)$$x^3 + x^2 + x + 1$$
ii)$$x^4 + x^3 + x^2 + x + 1$$
iii)$$x^4 + 3x^3 + 3x^2 + x + 1$$
iv)$$x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}$$

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

ii)Let $$p(x) = x^4 + x^3 + x^2 + x + 1$$
So, To check, whether x + 1 is a factor of $$x^4 + x^3 + x^2 + x + 1$$.
By factor theorem,
$$p(–1) = {-1}^4 + {–1}^3 + {–1}^2 + {-1} + 1$$
i.e. $$p(–1) = 1 – 1 + 1 - 1 + 1$$
i.e.$$p(-1) = 1$$
Hence, x + 1 is not the factor of $$x^4 + x^3 + x^2 + x + 1$$

iii)Let $$p(x) = x^4 + 3x^3 + 3x^2 + x + 1$$
So, To check, whether x + 1 is a factor of $$x^4 + 3x^3 + 3x^2 + x + 1$$.
By factor theorem,
$$p(–1) = {-1}^4 + 3{–1}^3 + 3{–1}^2 + {-1} + 1$$
i.e. $$p(–1) = 1 – 3 + 3 - 1 + 1$$
i.e.$$p(-1) = 1$$
Hence, x + 1 is not the factor of $$x^4 + 3x^3 + 3x^2 + x + 1$$

iv)Let $$p(x) = x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}$$
So, To check, whether x + 1 is a factor of $$x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}$$.
By factor theorem,
$$p(–1) = {–1}^3 - {–1}^2 - (2 + \sqrt{2}){-1} + \sqrt{2}$$
i.e. $$p(–1) = – 1 - 1 + 2 + \sqrt{2} + \sqrt{2}$$
i.e.$$p(-1) = 2\sqrt{2}$$
Hence, x + 1 is not the factor of $$x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}$$