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# The value of $m$, in order that ${x^2} - mx - 2$ is the quotient when ${x^3} + 3{x^2} - 4$ is divided by $x + 2$, isA. $-1$B. 1C. 0D. $-2$

Last updated date: 22nd Feb 2024
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Hint: Here we need to find the value of the variable. Here we have been given the quotient, dividend and quotient. So we will use the remainder theorem which states that the dividend is equal to addition of the product of quotient and divisor and the remainder. So we will substitute all the given values in this formula to get the value of the required variable.

Complete step by step solution:
It is given that when we divide ${x^3} + 3{x^2} - 4$ by $x + 2$, we get the quotient as ${x^2} - mx - 2$ and remainder as zero as it completely divides the dividend.
But here we need to calculate the value of $m$.
So here we will use the remainder theorem which states that the dividend is equal to addition of the product of quotient and divisor and the remainder.
Using this theorem, we get
$\Rightarrow {x^3} + 3{x^2} - 4 = \left( {{x^2} - mx - 2} \right)\left( {x + 2} \right) + 0$
On multiplying the terms in right side of the equation using the distributive property of multiplication, we get
$\Rightarrow {x^3} + 3{x^2} - 4 = {x^2}\left( {x + 2} \right) - mx\left( {x + 2} \right) - 2\left( {x + 2} \right)$
On multiplying the terms using the distributive property of multiplication, we get
$\Rightarrow {x^3} + 3{x^2} - 4 = {x^3} + 2{x^2} - m{x^2} - 2mx - 2x - 4$
On adding and subtracting the like terms, we get
$\Rightarrow {x^3} + 3{x^2} - 4 = {x^3} + \left( {2 - m} \right){x^2} - \left( {2m + 2} \right)x - 4$
On comparing the coefficients of the cubic equations of both sides, we get
$\Rightarrow 2 - m = 3$
On further simplification, we get
$\Rightarrow m = 2 - 3 = - 1$

Hence, the value of $m$ is $- 1$.
Thus, the correct answer is option A.

Note:
We have used the distributive property of multiplication to simplify the expression. It states that if $a$ , $b$ and $c$ are any numbers then according to this property, $\left( {a + b} \right)c = a \cdot c + b \cdot c$. In order to solve this question, we need to know some basic mathematical operations, such as multiplication, division, etc. The addition is the operation in which two numbers are combined to get the result. Subtraction is the operation which gives us the difference between the two numbers. Multiplication is the operation in which one number is added to itself for some particular number of times. The division is the operation in which the dividend is divided by the divisor to get the quotient along with some remainder where the dividend is the term or number which is to be divided.