($-$)  ($-$)

  ($+$)  ($+$)

Solution : We first arrange the terms of the dividend and the divisor in the decreasing order of their degrees. Recall that arranging the terms in this order is called writing the polynomials in standard form. In this example, the dividend is already in standard form, and the divisor, 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,

${(x^2 + 2x + 1) × (3x – 5) + (9x + 10) = 3x^3 + 6x^2 + 3x – 5x^2 – 10x – 5 + 9x + 10}$

${= 3x^3 + x^2 + 2x + 5 }$

		Here again, we see that
		Dividend =  Divisor × Quotient + Remainder 
		
		This says that

If ${p(x)}$ and ${g(x)}$ are any two polynomials with ${g(x) ≠ 0}$  ,
then we can find polynomials ${q(x)}$ and ${r(x)}$ such that
${p(x)  =  g(x) ×  q(x) + r(x),}$

where ${r(x) = 0}$ or degree of ${r(x) < }$ degree of ${g(x)}$.

This result is known as the Division Algorithm for polynomials.