Question

What must be added ${x^3} - 3{x^2} - 12x + 19$ so that the result is exactly divisible by ${x^2} + x - 6$?
A) $2x - 5$
B) $2x + 5$
C) $ - 2x - 5$
D) $x + 5$

Answer 1

Hint: In mathematics, the polynomial is an expression of two or more algebraic terms that contain different powers of the same variables. In algebra, polynomial long division is an algorithm for dividing a polynomial of the same degree or lower degree. Or using the factorization of polynomials, we can solve this problem.

Complete step-by-step solution:
In this problem, we solve this by using the factorization of the polynomial method.
Using the given polynomials, we can solve it.
Let $p\left( x \right) = {x^3} - 3{x^2} - 12x + 19$
and $q\left( x \right) = {x^2} + x - 6$ are the given polynomials in the problem.
$p\left( x \right)$ is divided by $q\left( x \right)$ and we get the remainder is $r\left( x \right)$.
Let, $r\left( x \right) = ax + b$ is a linear equation that is added to the given polynomial $p\left( x \right) = {x^3} - 3{x^2} - 12x + 19$ and is divisible by $q\left( x \right) = {x^2} + x - 6$.
Now let us take, $f\left( x \right) = p\left( x \right) + r\left( x \right)$
Now apply the given values in the equation. We have,
$f\left( x \right) = {x^3} - 3{x^2} - 12x + 19 + ax + b$
Now simplify this, we get
$f\left( x \right) = {x^3} - 3{x^2} + \left( {a - 12} \right)x + \left( {19 + b} \right)$ …………………………………………….$(1)$
In the above step, the common terms are collected.
Now we have,
$q\left( x \right) = {x^2} + x - 6$
Now factorize the term, we get
$q\left( x \right) = \left( {x + 3} \right)\left( {x - 2} \right)$
Here, $\left( {x + 3} \right)$ and $\left( {x - 2} \right)$ are the factors.
Therefore $x = - 3$ and $x = 2$
Now apply the values, we get
$f\left( { - 3} \right) = {\left( { - 3} \right)^3} - 3\left( { - 3} \right){}^2 + \left( {a - 12} \right)\left( { - 3} \right) + \left( {19 + b} \right)$
Now simplify this we get,
$ = - 27 - 27 - 3a + 36 + 19 + b$
Now solve this we get,
$ = - 3a + b - 54 + 55$
We get, $f\left( { - 3} \right) = - 3a + b + 1 = 0$;…………………………………………………..$\left( 2 \right)$
Now we have are going to find,
$f\left( 2 \right) = {\left( 2 \right)^3} - 2\left( 2 \right){}^2 + \left( {a - 12} \right)\left( 2 \right) + \left( {19 + b} \right)$
Now simplify this we get,
$ = 8 - 12 + 2a - 24 + 19 + b$
$ = 2a + b + 27 - 36$
Now simplify this we get,
$f\left( 2 \right) = 2a + b - 9 = 0$
$2a + b = 9$………………………………………………………………………………………………..$\left( 3 \right)$
Now solving the equations two and three we get,
$ - 3a + b = - 1 \\
  \Rightarrow 2a + b = 9 $
We get,
$\Rightarrow - 5a = - 10$
$\Rightarrow 5a = 10$
Solving this we get,
$\Rightarrow a = 2$
Now applying the value $a = 2$ we get,
$2a + b = 9$
$\Rightarrow 2\left( 2 \right) + b = 9$
$\Rightarrow 4 + b = 9$
$\Rightarrow b = 5$
Now we find the value of $a,b$
Now apply the value in the $r\left( x \right) = ax + b$, we get the required term.
$ = 2x + 5$
Hence the required term is $2x + 5$
Hence the answer is option A.
The explanation for the option B:
 By using the method of factorization of polynomials we prove that option A is the answer.
Therefore option B is not an answer.
The explanation for the option C:
From the above steps, we solved the problem and showed option A as an answer for the given problem.
Therefore option C is not a solution for the question.
The explanation for the option D:
We proved that option A is an answer for the given problem.
Therefore option D is not an answer.

Note: The above solution is solved by using the method of factorization of polynomials. This problem also can be solved by the method of long division. These two methods are appropriate methods to solve this problem.