Question

If the polynomial \[6{x^4} + 8{x^3} + 17{x^2} + 21x + 7\] is divided by another polynomial \[3{x^2} + 4x + 1\] the remainder comes out to be \[ax + b\]. Find \[a\] and \[b\].
A) \[0,1\]
B) \[1,2\]
C) \[1, - 1\]
D) \[ - 1,2\]

Answer 1

Hint:
Here, we will use the method of long division and divide the given Dividend by the Divisor to get the required remainder. Comparing the remainder by \[ax + b\], we will get the required values of \[a\] and \[b\] respectively.

Complete step by step solution:
We will use the method of long division to find the required remainder.
According to the question, Dividend i.e. the number which we are required to divide is \[6{x^4} + 8{x^3} + 17{x^2} + 21x + 7\]
And the Divisor, i.e. the number by which the above number is to be divided is \[3{x^2} + 4x + 1\].
Hence, by using long division method:

Hence, the quotient is \[2{x^2} + 5\] and the remainder is \[x + 2\]
Now, according to the question, we are given that the remainder comes out to be \[ax + b\].
So, writing the given remainder in \[ax + b\] form, we get
\[x + 2 = \left( 1 \right)x + \left( 2 \right)\]
Now comparing them together he two expression, we get
\[\left( 1 \right)x + \left( 2 \right) = ax + b\]
We can see that, \[a = 1\] and \[b = 2\]
Therefore, the required values of \[a\] and \[b\] are \[1,2\] respectively

Hence, option B is the correct answer.

Note:
We will check whether we have found the correct quotient and remainder or not by using the formula:
Dividend \[ = \] Divisor \[ \times \] Quotient \[ + \] Remainder
Substituting the values in this equation,
\[ \Rightarrow 6{x^4} + 8{x^3} + 17{x^2} + 21x + 7 = \left( {3{x^2} + 4x + 1} \right)\left( {2{x^2} + 5} \right) + x + 2\]
Now, solving for RHS by opening the brackets,
RHS \[ = 3{x^2}\left( {2{x^2} + 5} \right) + 4x\left( {2{x^2} + 5} \right) + 1\left( {2{x^2} + 5} \right) + x + 2\]
\[ \Rightarrow \] RHS \[ = 6{x^4} + 15{x^2} + 8{x^3} + 20x + 2{x^2} + 5 + x + 2\]
Hence, solving further, we get,
RHS \[ = 6{x^4} + 8{x^3} + 17{x^2} + 21x + 7\]
Clearly, RHS \[ = \] LHS (which is the dividend)
Hence, we have verified that our answer is correct.
Therefore, we can always check for our answer using this formula.