By divisible algorithm, when `p(x) = x^3 - 6x^2 - 15x + 80` is divided by `x^2 + x -12` the reminder is a linear polynomial
Let r(x) = a(x) + b be subtracted from p(x) so that the result is divisible by q(x).
Let
`f(x) = p(x) -q(x)`
` = x^3 - 6x^2 - 15x + 80 - (ax + b)`
` = x^3 -6x^2 - (a+15)x + 80 -b`
We have,
`q(x) = x^2 + x - 12`
` = x^2 + 4x -3x -12`
` = (x+4)(x-3)`
Clearly, (x+4)and (x-3)are factors of q(x), therefore, f(x) will be divisible by q(x) if (x+4) and (x-3) are factors of f(x), i.e. f (−4) and f (3) are equal to zero.
Therefore,
`f(-4) = (-4)^3 -6(-4)^2 - (a+15)(-4) + 80 -b = 0`
`-64 -96 + 4a + 60 + 80 -b = 0`
` -20 +4a -b = 0`
`+4a -b = 20`
and
`f(3) = (3)^2 - (a+15)(3) + 80 -b = 0`
`27 - 25 - 3a - 45 + 80 -b =0`
` -3a - b = 8`
`3a +b = 8`
Adding (i) and (ii), we get,
a=4
Putting this value in equation (i), we get,
b=-4
Hence, `x^3 - 6x^2 - 15x + 80` will be divisible by `x^2 + x - 12,` if 4 x − 4 is subtracted from it.
Page 2
By division algorithm, when p(x) = 3x3 + x2 − 22x + 9 is divided by `3x^2 + 7x - 6,`the reminder is a linear polynomial. So, let r(x) = ax + b be added to p(x) so that the result is divisible by q(x)
Let
`f(x) = p(x) + r(x)`
` = 3x^2 + x^2 - 22x + 9 ax +b`
` = 3x^2 + x^2 +(a- 22) x + 9 + b`
We have
\[q\left( x \right) = 3 x^2 + 7x - 6\]
\[ = 3 x^2 + 9x - 2x - 6\]
\[ = 3x\left( x + 3 \right) - 2\left( x + 3 \right)\]
\[ = \left( 3x - 2 \right) \left( x + 3 \right)\]
Clearly,
\[\left( 3x - 2 \right)\] and \[\left( x + 3 \right)\]
are factors of q(x).
Therefore, f(x) will be divisible by q(x) if (3x - 2)and (x + 3)are factors of f(x), i.e.,
`f (2/3)`and f(−3) are equal to zero.
Now,
\[f\left( \frac{2}{3} \right) = 0\]
\[ \Rightarrow 3 \left( \frac{2}{3} \right)^3 + \left( \frac{2}{3} \right)^2 + \left( a - 22 \right)\left( \frac{2}{3} \right) + 9 + b = 0\]
\[ \Rightarrow 3 \times \frac{8}{27} + \frac{4}{9} + \frac{2a}{3} - \frac{44}{3} + 9 + b = 0\]
\[ \Rightarrow \frac{8}{9} + \frac{4}{9} - \frac{44}{3} + 9 + \frac{2a}{3} + b = 0\]
\[ \Rightarrow \frac{8 + 4 - 132 + 81}{9} + \frac{2a}{3} + b = 0\]
\[ \Rightarrow - \frac{39}{9} + \frac{2a}{3} + b = 0\]
\[ \Rightarrow \frac{2a}{3} + b = \frac{13}{3}\]
\[ \Rightarrow 2a + 3b = 13 . . . . . . . . \left( i \right)\]
And
\[f\left( - 3 \right) = 0\]
\[ \Rightarrow 3 \left( - 3 \right)^3 + \left( - 3 \right)^2 + \left( a - 22 \right)\left( - 3 \right) + 9 + b = 0\]
\[ \Rightarrow - 81 + 9 - 3a + 66 + 9 + b = 0\]
\[ \Rightarrow - 3a + b = - 3 \]
\[ \Rightarrow b = - 3 + 3a . . . . . . . . . \left( ii \right)\]
Substituting the value of b from (ii) in (i), we get,
\[2a + 3\left( 3a - 3 \right) = 13\]
\[ \Rightarrow 2a + 9a - 9 = 13\]
\[ \Rightarrow 11a = 13 + 9\]
\[ \Rightarrow 11a = 22\]
\[ \Rightarrow a = 2\]
Now, from (ii), we get
\[b = - 3 + 3\left( 2 \right) = - 3 + 6 = 3\]
So, we have a = 2 and b = 3
Hence, p(x) is divisible by q(x), if 2x + 3is added to it.