**Divide polynomials using long division :**

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

Let us see some examples based on the above concept.

**Example 1 :**

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

**Example 2 :**

Find the quotient and the remainder when (4x³ + 6x² - 23 x - 15) is divided by 3 + x

**Solution :**

**Step 1 :**

Divide the first term of the dividend by the first term of the divisor. That is 4x³/x = 4x². Now we have to multiply this 4x² by the divisor. So we get 4x³ + 12x².

In order to subtract 4x³ + 12x², we changed the sign. After simplifying we get - 6x² - 23x - 15.

**Step 2 :**

Divide the first term of the dividend by the first term of the divisor. That is -6x²/x = -6x. Now we have to multiply this -6x by the divisor(x + 3). So we get -6x² - 18 x.

Subtract -6x² - 18 x from -6x² - 23 x - 15. By subtracting these two polynomials we get -5x - 15.

**Step 3 :**

-5x/x = -5. Now we have to multiply this -5 by the divisor(x + 3). So we get -5x - 15.

By subtracting these two polynomials, we get 0.

Hence the quotient = 4 x²-6 x - 5 and the remainder = 0.

After having gone through the stuff given above, we hope that the students would have understood "Divide polynomials using long division".

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