Note that we stop the division process when either the
remainder is zero or its degree is less than the degree of the divisor. So,
here the quotient is 2x – 1 and the remainder is 3.
Therefore, Dividend = Divisor × Quotient + Remainder
Let us now extend this process to divide a polynomial by a
quadratic polynomial.
Example-9.
Divide 3x
3
+ x
2
+ 2x + 5 by 1 + 2x + x2.
Solution :
We first arrange the terms of the dividend
and the divisor in the decreasing order of their degrees.
(Arranging the terms in this order is termed as writing
the polynomials in its standard form). In this example,
the dividend is already in its standard form, and the divisor
is also in standard form, is x
2 + 2x + 1.
Step 1 : To obtain the first term of the quotient, divide
the highest degree term of the dividend (i.e., 3x
3
) by the
highest degree term of the divisor (i.e., x
2
). This is 3x.
Then carry out the division process. What remains is –5x
2
–x+5.
Step 2 : Now, to obtain the second term of the quotient, divide the highest degree term of the
new dividend (i.e., – 5x
2
) by the highest degree term of the divisor (i.e., x
2
). This gives – 5. Again
carry out the division process with– 5x
2
– x + 5.
Step 3 : What remains is 9x + 10. Now, the degree of 9x + 10 is less than the degree of the
divisor x
2
+ 2x + 1. So, we cannot continue the division any further.
So, the quotient is 3x – 5 and the remainder is 9x + 10. Also,
the dividend is already in its standard form, and the divisor
is also in standard form, is x
2 + 2x + 1.