Question

A village of North-East in India is suffering from a flood. A merchant from Mumbai decides to help them with food items, clothes etc. so, he collects some money from different persons, which is represented by \[{x^4} + {x^3} + 8{x^2} + ax + b\] . If the number of persons contributed for his service are \[{x^2} + 1\] , find the value of \[a\] and \[b\] .

Answer 1

Hint: Here we have to consider the number of persons as a quadratic function. Becomes a merchant collecting money from different persons. Then using the division algorithm and comparing the coefficients of like powers of \[x\] we find the values of the coefficient and the values of \[a\] and \[b\] .

Complete step by step solution:
In the given information, given the total amount as a function of \[x\]
i.e., \[{x^4} + {x^3} + 8{x^2} + ax + b\] . --(1)
Also given that each person's service is \[{x^2} + 1\] .
Suppose that the number of persons are \[{x^2} + cx + d\] (we cannot take linear as given that a merchant collected money from different persons).
Then applying the division algorithm. We get
 \[{x^4} + {x^3} + 8{x^2} + ax + b = \left( {{x^2} + 1} \right)\left( {{x^2} + cx + d} \right)\]
 \[ \Rightarrow {x^4} + {x^3} + 8{x^2} + ax + b = {x^4} + c{x^3} + (d + 1){x^2} + cx + d\] ---(2)
Then, equating the coefficients of the like powers of \[x\] in the equation (2), we get
 \[c = 1\] , \[d + 1 = 8\] , \[c = a\] and \[d = b\] .
 \[ \Rightarrow c = 1\] , \[a = 1\] , \[d = 7\] and \[b = 7\] .
Hence, the number of persons are \[{x^2} + x + 7\] .
The values of \[a\] and \[b\] are \[1\] and \[7\] respectively.
So, the correct answer is “a = 1 and b = 7”.

Note: Note that the division algorithm states that: If \[a\] and \[b\] are any two positive integers such that \[a > b\] . Then there exists two positive-integers \[r\] and t such that \[a = bt + r,0 \leqslant r < b\] . Here \[r\] is called the remainder.