Question

(x - 1) and (x + 2) are the factors of x³ + 10x² + px + q. Find p and q.

Answer 1

By factor theorem, if (x - a) is a factor of a polynomial P(x) in x, then P(a) = reminder of P(x) when divided by (x - a) = 0.

$P(1) = 1^3 + 10*1^2 + p * 1 + q = 0 \\ \\ 1 + 10 + p + q = 0 \\ \\ p + q = -11 \\ \\ \\ P(-2) = -2^3 + 10*(-2)^2 + p(-2) + q = 0 \\ \\ -8 + 40 -2 p + q = 0 \\ \\ -2 p + q = -32 \\ \\ 2p - q = 32 \\ \\ Add\ the\ two\ equations, \\ \\ p+q+2p -q = -11+32 \\ \\ 3p = 21 \\ \\ p = 7 \\ \\ substitute\ this\ in\ p+q = -11,\ to\ get\ \\ \\ q = -18$

Answer 2

Let,if p(x) is a factor of (x-1) and (x-2) then remainder is p(1)=0 and p(2)=0.
P(1)=1+10+p+q
          11+p+q=0-----(1)
P(-2)=-8+10(4)+(-2)p+q
           -8+40-2p+q=0
              32-2p+q=0..........(2)
Adding eq:1 and 2,
11+p+q=0
32-2p+q=0
-      +      -
-21+3p=0
3p=21
P=7
Substitute p value in equation 1
11+p+q=0
11+7+q=0
18+q=0
q=-18




P=7 and q=-18