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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,
$$\therefore p(–1) = 2{–1}^3 + {–1}^2 - 2{-1} - 1$$
$$\Rightarrow p(–1) = –2 + 1 + 2 - 1$$
$$\Rightarrow p(-1) = 0$$

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,
$$\therefore p(–2) = (–2)^3 + 3(–2)^2 + 3(–2) + 1$$
$$\Rightarrow p(–2) = –8 + 12 - 6 + 1$$
$$\Rightarrow p(-2) = -1$$

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,

$$\therefore p(3) = (3)^3 - 4(3)^2 + (3) + 6$$
$$\Rightarrow$$ $$p(3) = 27 - 36 + 3 + 6$$
$$\Rightarrow$$ $$p(3) = 0$$

Hence, $$g(x) = x - 3$$ is the factor of $$p(x) = x^3 - 4x^2 + x + 6$$