Question

lets see how fast I get answers

Answer 1

1)p(3)=(3)^3-4(3)^2+3+6
=27-36+9
=0
hence it is a factor
2)p(-2)=(-2)^3+3(-2)^2+3(-2)+1
=-8+12-6+1
=-14+13
=-1
hence it is not a factor

Answer 2

(i)

Given P(x) = x^3 - 4x^2 + x + 6.

Given g(x) = x - 3.

For x - 3 to be a factor of P(x), the factor theorem says that x = 3 must be a zero of P(x).

⇒ x - 3 = 0

⇒ x = 3.

Plug x = 3 in P(x), we get

⇒ P(3) = (3)^3 - 4(3)^2 + 3 + 6

           = 27 - 36 + 3 + 6

           = -9 + 9

           = 0.

Since, remainder is 0. Therefore, x - 3 is a factor of x^3 - 4x^2 + x + 6.

-----------------------------------------------------------------------------------------------

(ii)

Given P(x) = x^3 + 3x^2 + 3x + 1.

Given g(x) = x + 2.

If x + 2 is a factor, then x = -2 is a root.

Plug x = -2 in P(x), we get

⇒ P(-2) = (-2)^3 + 3(-2)^2 + 3(-2) + 1

            = -8 + 12 - 6 + 1

            = 4 - 6 + 1

            = -2 + 1

            = -1.

∴ Remainder is not 0. Therefore, x + 2 is not a factor of x^3 + 3x^2 + 3x + 1.

Hope this helps!