Question

The value of \[m\], in order that \[{x^2} - mx - 2\] is the quotient when \[{x^3} + 3{x^2} - 4\] is divided by \[x + 2\], is
(a) \[ - 1\]
(b) 1
(c) 0
(d) \[ - 2\]

Answer 1

Hint: Here, we have to find the value of \[m\]. We will use a long division method to divide \[{x^3} + 3{x^2} - 4\] by \[x + 2\] and obtain the quotient. Then, we will compare the coefficients of the two quotients to find the value of \[m\].

Complete step-by-step answer:
Let us use a long division method to divide \[{x^3} + 3{x^2} - 4\] by \[x + 2\].

We can observe that when \[{x^3} + 3{x^2} - 4\] is divided by \[x + 2\] using a long division method, the quotient is \[{x^2} + x - 2\].
It is given that \[{x^2} - mx - 2\] is the quotient when \[{x^3} + 3{x^2} - 4\] is divided by \[x + 2\].
Thus, we will compare the two quotients.
Comparing the coefficients in the quotients \[{x^2} - mx - 2\] and \[{x^2} + x - 2\], we get
\[ - m = 1\]
Multiplying both the sides by \[ - 1\], we get
\[ \Rightarrow - m\left( { - 1} \right) = 1\left( { - 1} \right)\]
Therefore, we get
\[ \Rightarrow m = - 1\]
Thus, we found out that the value of \[m\] is \[ - 1\].
\[\therefore \] The correct option is option (a).

Note: We need to be careful while doing the long division. When multiplying \[x + 2\] by \[x\], the product of \[2\] and \[x\] is \[2x\]. Since the dividend has no term from which \[2x\] can be subtracted, we have placed the term at the end of the dividend.
If \[g\left( x \right)\] is a factor of a polynomial \[p\left( x \right)\], then \[p\left( x \right)\] is exactly divisible by \[g\left( x \right)\]. This means that when \[p\left( x \right)\] is divided by \[g\left( x \right)\], the remainder should be equal to 0.
We can see that when \[{x^3} + 3{x^2} - 4\] is divided by \[x + 2\], the remainder is 0.
Therefore, the polynomial \[x + 2\] is a factor of the polynomial \[{x^3} + 3{x^2} - 4\].