Which of the following is not a possible rational root of the polynomial below

Question

f(x)= 5x^3+2x^2-x-20 A) 1/4 B) 4 C) 4/5 D) 1?

Answer 1

NOTE : if f (x ) = 0,  then x is a rational root to the polynomial f (x).

The polynomail is f ( x ) = 5x ^3 + 2x ^2 - x - 20.

• If x = 1/4 then,

f ( 1/4 ) = 5( 1/4 )^3 + 2( 1/4 )^2 - ( 1/4 ) - 20

           = ( 5/64 ) + ( 2/16 ) - ( 1/4 ) - 20

           = [ 5 + 8 - 16 - 1280 ] / 64

           = - 1283/64.

- 1283/64 ≠ 0, so, 1/4 is not a possible rational root of 5x ^3 + 2x ^2 - x - 20.

• If x = 4 then,

f ( 4 ) = 5( 4 )^3 + 2( 4 )^2 - ( 4 ) - 20

           = ( 5*64 ) + 32 - 4 - 20

           = 320 + 8

           = 328.

328 ≠ 0, so, 4 is not a possible rational root of 5x ^3 + 2x ^2 - x - 20.

• if x = 4/5 then,
  

  f ( 4 ) = 5( 4/5 )^3 + 2( 4/5 )^2 - ( 4/5 ) - 20

             = ( 64/25 ) + (32/25) - ( 4/5 ) - 20

             = ( 64 + 32 - 20 - 500 )/25

             = - 424/25.

- 424/25 ≠ 0, so, 4/5 is not a possible rational root of 5x ^3 + 2x ^2 - x - 20.

• If x = 1 then,

f ( 4 ) = 5( 1 )^3 + 2( 1 )^2 - ( 1 ) - 20

           = 5 + 2 - 1 - 20

           = 7 - 21

           = - 14.

- 14 ≠ 0, so, 4 is not a possible rational root of 5x ^3 + 2x ^2 - x - 20.

Given options are not a possible rational roots of the given polynomail f ( x ) = 5x ^3 + 2x ^2 - x - 20.