#### Mathematics (NTSE/Olympiad)

# Polynomials

__Working Rule to divide a Polynomial by Another Polynomial __

**Working Rule to divide a Polynomial by Another Polynomial **

**Step 1:** First arrange the term of dividend and the divisor in the decreasing order of their degrees.

**Step 2 :** To obtain the first term of quotient divide the highest degree term of the dividend by the highest degree term of the divisor.

**Step 3 :** To obtain the second term of the quotient, divide the highest degree term of the new dividend obtained as remainder by the highest degree term of the divisor.

**Step 4 :** Continue this process till the degree of remainder is less than the degree of divisor.

**Division Algorithm for Polynomial **

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) = q(x)×g(x)+r(x)**

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

The result is called **Division Algorithm for polynomials**.

**DIVIDEND = QUOTIENT ****× DIVISOR +** **REMAINDER**

**Example: Divide 3x**^{3} + 16x^{2} + 21x + 20 by (x + 4).

Solution:

Quotient = 3x^{2} + 4x + 5 and Remainder = 0

**Example: Apply the division algorithm to find the quotient and remainder on dividing p(x) by g(x) as given : ****p(x) = x**^{3} – 3x^{2} + 5x – 3, g(x) = x^{2} – 2

Solution: We have,

p(x) = x^{3} – 3x^{2} + 5x – 3 and g(x) = x^{2} – 2

We stop here since degree of (7x – 9) < degree of (x^{2} – 2). So, quotient = x – 3, remainder = 7x – 9, Therefore,

**Quotient × Divisor + Remainder**

= (x – 3) (x^{2} – 2) + (7x – 9)

= x^{3} – 2x – 3x^{2} + 6 + 7x – 9

= x^{3} – 3x^{2} + 5x – 3 = Dividend

Therefore, the division algorithm is verified.