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`

Use the Factor Theorem to determine whether g(x) is a factor of p(x) i

1.	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`
Answer» From Factor theorem, `(y-a)` is a factor of `P(y) if P(a) = 0.` (i)`p(x) = 2x^3+x^2-2x-1 and g(x) = x+1` For `g(x)` to be a factor of `p(x)`, `p(-1)` should be `0`. `p(-1) = 2(-1)^3+(-1)^2-2(-1) -1 ` `=-2+1+2-1 =0`As, `p(-1)` is `0`, `(x+1)` is a factor of `p(x)`. (ii) `p(x) = x^3+3x^2+3x+1 and g(x) = x+2` For `g(x)` to be a factor of `p(x)`, `p(-2)` should be `0`. `p(-2) = (-2)^3+3(-2)^2+3(-2)+1` `=-8+12-6+1 = -1` As, `p(-2)` is not equal to `0`, `(x+2)` is not a factor of `p(x)`. (iii) `p(x) = x^3+4x^2+x+6 and g(x) = x-3` For `g(x)` to be a factor of `p(x)`, `p(-3)` should be `0`. `p(3) = (3)^3+4(3)^2+3+6` `=27+36+3+6=72` As, `p(3)` is not equal to `0`, `(x-3)` is not a factor of `p(x)`.