3. Square root by long division method

Find the square root of \(9x^4 + 12x^3 + 10x^2 + 4x + 1\).

 

Solution:

 

Step 1: Group the terms as a pair from the unit place. That is, \(9x^4 + \overline{12x^3 + 10x^2} + \overline{4x + 1}\).

 

Step 2: Here, the extreme left number is \(9x^4\). We must select the number whose square is equal to or less than the given number. Here, the square of the square of the number is \(3x^2\).

 

Thus, we have:

 

 

Step 3: Subtracting the numbers, we get \(0\), and bring the next pair to the right of the remainder.

 

 

Step 4: Find the new divisor by multiplying the quotient by \(2\). And, the next digit can be selected which is less than or equal to the given term \(12x^3 + 10x^2\).

 

 

Step 5: Subtract the numbers and write the result, and bring the next pair to the right of the remainder.

 

 

Step 6: Find the new divisor by multiplying the quotient by \(2\). And, the next digit can be selected which is equal to the given term \(6x^2 + 4x + 1\).

Therefore, \(\sqrt{9x^4 + 12x^3 + 10x^2 + 4x + 1} = |3x^2 + 2x + 1|\).