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 \(p(x) = x^3 - 4x^2 + x + 6\)