+0

0
241
2
+159

a is a positive integer

equation: x^3 + (a+17)x^2 + (38-a)x - 56 = 0

The equation has 3 integer roots

what is the value of a?

what are the three roots?

hint: x=1 is one of the roots

Oct 25, 2018

#1
+103049
+2

If 1 is a root...let's perform  some synthetic division

1  [  1     a + 17    38   - a      - 56  ]

1      a + 18        56

_________________________

1     a + 18        56            0

The remaining polynomial is

x^2  + (a + 18)x  +  56

Factors of 56  are  1   2   4    7   8   14   28   56

Since a  is positive...only the pairs  2, 28    and  1, 56   will give us what we want

So...the possibiliites for the factorization of this are

(x +28) ( x + 2) =  x^2  + 30x  + 56       so  a  =  12       or

( x + 56) ( x + 1)  = x^2  + 57x  +  56    so   a  = 39

Checking this

x^3 + 29x^2 + 26x  - 56  = 0  produces the three roots of 1, -28  and  -2

And

x^3  + 56x^2 -1x - 56  = 0   produces the three roots   1, -1, - 56

Oct 25, 2018
#2
+159
+1

thank you!!

hearts123  Oct 27, 2018