Question

Factor each polynomial completely if it is prime write prime

11. +3+ 4+2 + 3t + 12 12. 822 + 10z-3

Answer 1

Given

t^3 + 4*t^2 + 3*t + 12

We have to check for what value of t , this polynomial becomes zero

Put t = - 4

=> ( - 4 )^3 + 4*( - 4 )^2 + 3*( - 4 ) + 12

=> - 64 + 4*16 - 12 + 12

=> - 64 + 64 - 12 + 12

=> 0

So, ( t + 4 ) is a factor of the polynomial

=> t^3 + 4*t^2 + 3*t + 12 = t^2 * ( t + 4 ) + 3 * ( t + 4 ) + 0 * ( t + 4 )

=> t^3 + 4*t^2 + 3*t + 12 = ( t + 4 )*( t^2 + 3 )

Now ( t^2 + 3 ) cannot be factored further

So, the polynomial t^3 + 4*t^2 + 3*t + 12 is Prime

Given

8*z^2 + 10*z - 3

=> 8*z^2 + 12*z - 2*z - 3

=> - 2*z*( - 4*z + 1 ) - 3*( - 4*z + 1 )

=> ( - 2*z - 3 )*( - 4*z + 1 )

=> ( - 1 )*( 2*z + 3 )*( - 4*z + 1 )

=> ( 2*z + 3 )( 4*z - 1 )

So, polynomial 8*z^2 + 10*z - 3 can be factored into  ( 2*z + 3 )( 4*z - 1 )