If (x-2) and (x-1) are factors of polynomial `x^3+px^2+qx+1`  what is the sum of p and q.

Expert Answers
durbanville eNotes educator| Certified Educator

You can confirm your answer if required by finding actual values for p and q using both equations as follows : 

 Using factor `(x-2) ` the sum of p+q will be :

`f(2) = 0 = x^3+px^2+qx+1`

`therefore 0=(2)^3+p(2)^2+q(2)+1`

`therefore 0= 8+4p+2q+1`

`therefore 0= 9 +4p+2q`

Now solve simultaneously using both factors `(x-1)(x-2)` :

`0=9+4p +2q` and

`p+q=-2` `therefore p +q+2=0`

Multiply this equation by 2 so that we can eliminate q (there are various ways to solve) when we combine the equations:


As they both equal zero make them equal to each other:

`therefore 9+4p+2q=2p+2q+4`

`therefore 9+4p+2q-2p-2q-4=0`

`therefore 5+2p=0` (Note that q cancelled out)

`therefore 2p= -5`

`therefore p = -5/2`

Now solve for q using a previous equation and substituting for p:


`therefore -5/2+q=-2`

`therefore q= -2+5/2`

`therefore q=1/2` ``

`therefore p+q=-5/2+1/2`

`therefore p+q=-2`



durbanville eNotes educator| Certified Educator

Please note the error: The factors are `(x-2) (x-1)`

`therefore x=2 or x=1`

As we have : `x^3 + px^2 + qx+1` we can create equations using substitution:

`therefore f(1) = (1)^3 + p(1)^2 +q(1) +1`

`therefore = 1+p + q +1`

As `(x-1)`  is a factor it renders it equal to zero:

`therefore 0=` `1+p+q+1`

`therefore p+q = 0-1-1`

`therefore p+q=-2`

Ans : the sum of p and q is -2