Let p (x) = x3-3x2-12x+19 and q (x) = x2+x-6
By division algorithm, when p (x) is divided by q (x), the remainder is a linear expression in x.
So, let r (x) = ax + b is added to p (x) so that p (x) + r (x) is divisible by q (x).
Let,
f (x) = p (x) + r (x)
= x3 - 3x2 - 12x + 19 + ax + b
= x3 – 3x2 + x (a – 12) + b + 19
We have,
q (x) = x2+x-6
= (x + 3) (x – 2)
Clearly, q (x) is divisible by (x – 2) and (x + 3) i.e. (x – 2) and (x + 3) are factors of q (x)
We have,
f (x) is divisible by q (x)
(x – 2) and (x + 3) are factors of f (x)
From factor theorem,
If (x – 2) and (x + 3) are factors of f (x) then f (2) = 0 and f (-3) = 0 respectively.
f (2) = 0
(2)3 – 3 (2)2 + 2 (a – 12) + b + 19 = 0
⇒ 8 – 12 + 2a – 24 + b + 19 = 0
⇒ 2a + b – 9 = 0 (i)
Similarly,
f (-3) = 0
(-3)3 – 3 (-3)2 + (-3) (a – 12) + b + 19 = 0
⇒ -27 – 27 – 3a + 36 + b + 19 = 0
⇒ b – 3a + 1 = 0 (ii)
Subtract (i) from (ii), we get
b – 3a + 1 – (2a + b – 9) = 0 – 0
⇒ b – 3a + 1 – 2a – b + 9 = 0
⇒ - 5a + 10 = 0
⇒ 5a = 10
⇒ a = 2
Put a = 2 in (ii), we get
b – 3 × 2 + 1 = 0
⇒ b – 6 + 1 = 0
⇒ b – 5 = 0
⇒ b = 5
Therefore, r (x) = ax + b
= 2x + 5
Hence, x3 – 3x – 12x + 19 is divisible by x2 + x – 6 when 2x + 5 is added to it.
Let p(x) = x3 − 3x2 − 12x + 19 and `q(x) = x^2 + x -6 `be the given polynomial.
When p(x) is divided by q(x), the reminder is a linear polynomial in x.
So, let r(x) = ax + b is added to p(x), so that p(x) + r(x) is divisible by q(x).
Let f(x) = p(x) + r(x)
Then,
`f(x) = x^3 -3x^2 -12x + 19 +ax + b`
` = x^3 -3x^2 + (a-12)x+ (19+b)`
We have,
`q(x) = x^2 + x-6`
` = (x +3)(x - 2)`
Clearly, q(x) is divisible by (x+3)and (x- 2)i.e., (x+3) and (x-2) are the factors of q(x).
Therefore, f (x) is divisible by q(x), if (x + 3) and (x-2)are factors of f(x), i.e.,
f(-3)and f(2) = 0
Now, f(-3) = 0
\[\Rightarrow\] f(-3) = (-3)3 -3(-3)2 + (a-12)(-3)+19+b = 0
\[\Rightarrow\] -27 - 27 - 3a + 36 + 19 + b = 0
\[\Rightarrow\] -27 - 27 - 3a + 36 + 19 + b = 0\[\Rightarrow\] -54 - 3a + b + 55 = 0
\[\Rightarrow\] -3a + b + 1 = 0 ---- (i)
And
`f(2) = (2)^3 -3(2)^2 + (a-12) + 19 +b = 0`
`8-12+2a - 24 +19 +b = 0`
`2a +b = 9 ............. (2)`
Subtracting (i) from (ii), we get,
`(2a +b)-(-3a + b) = 10`
`5a = 10`
`a=2`
Putting this value in equation (ii), we get,
`⇒ 2 xx 2 +b = 9`
`b = 5`
Hence, p(x) is divisible by q(x) if (2x + 5)added to it.