Question

How do you determine the value of \[k\] if the remainder is \[3\] given \[\left( {{x^3} + k{x^2} + x + 5} \right) \div \left( {x + 2} \right)\]?

Answer 1

Hint: To solve the above question, we will use the remainder theorem. The remainder theorem states that: if we divide a polynomial \[f\left( x \right)\] by \[\left( {x - h} \right)\], then the remainder is \[f\left( h \right)\]. Here, we will put \[x = - 2\] in the given polynomial and whatever value we get we will equate that to the given remainder i.e., \[3\] to find the value of \[k\].

Complete step by step answer:
Let,
\[f\left( x \right) = {x^3} + k{x^2} + x + 5\]
\[\left( {x - h} \right) = x + 2\]
On comparing, we get;
\[ - h = 2\]
\[ \Rightarrow h = - 2\]
Now we will find the value of \[f\left( h \right)\].
We have;
\[f\left( x \right) = {x^3} + k{x^2} + x + 5\]
Putting \[x = h\] we get;
\[ \Rightarrow f\left( h \right) = {h^3} + k{h^2} + h + 5\]
Putting \[h = - 2\], we get;
\[ \Rightarrow f\left( { - 2} \right) = {\left( { - 2} \right)^3} + k{\left( { - 2} \right)^2} - 2 + 5\]
Solving we get;
\[ \Rightarrow f\left( { - 2} \right) = - 5 + 4k\]
According to the remainder theorem, this is the remainder. But the given remainder is \[3\]. So, we will equate it to \[3\]. So, we get;
\[ \Rightarrow - 5 + 4k = 3\]
On shifting we get;
\[ \Rightarrow 4k = 8\]
\[ \therefore k = 2\]

Note: One thing to note is that many students on seeing the question may think of using the long division method and then finding the remainder and then equating the remainder to the given remainder to find the value of k. Of course, we can use the long division method to solve the question and we will get the same answer but that process is too lengthy and takes a lot of time to solve and may even result in mistakes. So, it is easier to use the remainder theorem.