Question

If x³ – 2x² + px + q has a factor (x + 2) and leaves a remainder 9, when divided by (x + 1), find the values of p and q. With these values of p and q, factorize the given polynomial completely.

Answer 1

f(x) = x³ – 2x² + px + q
(x + 2) is a factor
f(-2) = (-2)³ – 2(-2)² + p (-2) + q
= –8 – 2 x 4 – 2p + q
= –8 – 8 – 2p + q
= –16 – 2p + q
∵ (x + 2) is a factor of f(x)
∴ f(–2) = 0
⇒ –16 – 2p + q = 0
⇒ 2p – q = –16       ....(i)
Again, let x + 1 = 0,
then x = –1
f(–1) = (–1)³ –2(–1)² + p(–1) + q
= –1 – 2 x 1 – p + q
= –1 – 2 – p + q
= –3 – p + q
∵ Remainder = 9, then
–3 –  p + q = 9
⇒ –p + q = 9 + 3 = 12
–p + q = 12         ....(ii)
Adding (i) and (ii)
p = –4
Substituting the value of p in (ii)
– (–4) + q = 12
4 + q = 12
⇒ q = 12 – 4 = 8
∴ p = –4, q = 8
∴ f(x) = x³ – 2x² – 4x + 8
Dividing f(x) by (x + 2), we get
f(x) = (x + 2)(x² – 4x + 4)
= (x + 2){(x)² – 2 x x(–2) + (2)²}
= (x + 2)(x – 2)² 
`x + 2")"overline(x^3 - 2x^2 - 4x + 8)("x^2 - 4x + 4`
           x³ + 2x²
          –    –                         
                –4x² – 4x
                –4x² – 8x
                +      +               
                   4x + 8
                   4x + 8
                –      –               
                       x