**Dividing polynomials using long division :**

The division of polynomials p(x) and g(x) is expressed by the following “division algorithm” of algebra.

Let p(x) and g(x) be two polynomials such that degree of p(x) ≥ degree of g(x) and g(x) ≠ 0. Then there exists unique polynomials q(x) and r (x)

such that

p(x) = g(x) q(x) + r (x) ... (1)

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

The polynomial p(x) is the dividend, g(x) is the divisor, q(x) is the quotient and r (x) is the remainder.

(1) ==> Dividend = (Divisor x Quotient) + Remainder

**Example 1 :**

**Divide the polynomial 2x**³ - 6x² + 5x + 4 by (x - 2)

**Solution : **

Let P(x) = 2x³ - 6x² + 5x + 4 and g(x) = x - 2

To divide the given polynomial by x - 2, we have divide the first term of the polynomial P(x) by the first term of the polynomial g(x).

If we divide 2x³ by x, we get 2x². Now we have to multiply this 2x² by x - 2. From this we get 2x³ - 4x².

Now we have to subtract 2x³ - 4x² from the given polynomial. So we get -2x² + 5x + 4.

Now we have to subtract 2x³ - 4x² from the given polynomial. So we get -2x² + 5x + 4.

repeat this process until we get the degree of p(x) ≥ degree of g(x)

Hence the quotient = 2x² - 2x + 1 and remainder = 6

**Example 2 :**

Find the quotient and remainder when 4x³ - 5x² + 6x - 2 by x - 1.

**Solution :**

Hence the quotient = 4x² - x + 5 and remainder = 3

**Example 3 :**

Find the quotient and remainder when x³ - 7x² - x + 6 by x + 2.

**Solution :**

Hence the quotient = x² - 9x + 17 and remainder = -28

**Example 4 :**

Find the quotient and the remainder when (10- 4x + 3x²) is divided by x - 2.

**Solution :**

Let us first write the terms of each polynomial in descending order ( or ascending order).

Thus, the given problem becomes (10- 4x + 3x²) ÷ (x - 2)

f(x) = 10- 4x + 3x²

= 3x² - 4x + 10

g (x) = x - 2

**Step 1 :**

In the first step, we are going to divide the first term of the dividend by the first first term of the divisor.

After changing the signs, +3x² and - 3x² will get canceled. By simplifying we get 2x + 10

**Step 2 :**

In the second step again we are going to divide the first term that is 2x by the first term of divisor that is x.

Quotient = 3x + 2

Remainder = 14

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