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Step-wise procedure for dividing a polynomial in one variable by a monomial in the same variable :

Step 1) Obtain the polynomial (dividend) and the monomial (divisor).

Step 2) Arrange the terms of the dividend in descending order of their degrees.

For example, write \(3x^2 – x^3 + 2x + 1 + 3x^3 – x^4\) as \(-x^4 – x^3 + 3x^3 + 3x^2 + 2x + 1\)

Step 3) Divide each term of the polynomial by the given monomial by using the rules of division of a monomial by a monomial i.e., **(a) Divide numbers (coefficients) and b) Subtract exponents.
**Step 4) Write the appropriate sign in between the terms.

The number of terms in the polynomial equals the number of terms in the answer when dividing by a monomial.

Numbers do not cancel and disappear. A number divided by itself is 1.

#### Division of a Polynomial by a Monomial Example1:

Divide (12x + 36) by 4.

**Solution:**

Here, the polynomial is (12x + 36) and the monomial is 4. So, by dividing (12x + 36) by 4 and by following rules of division of a monomial by a monomial, we get the result as 3x + 9.

#### Division of a Polynomial by a Monomial Example2:

Divide (\(3x^2 – x\)) by (-x).

**Solution:**

Here, the polynomial is (\(3x^2 – x\)) and the monomial is (-x). So, by dividing (\(3x^2 – x\)) by (-x) and by following rules of division of a monomial by a monomial, we get the result as -3x + 1.

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#### Division of a Polynomial by a Monomial Example3:

Divide (\(-12a^3b + 18a^2b^2 – 24ab^3\)) by (-6ab).

**Solution:**

Here, the polynomial is (\(-12a^3b + 18a^2b^2 – 24ab^3\)) and the monomial is (-6ab). So, by dividing (\(-12a^3b + 18a^2b^2 – 24ab^3\)) by (-6ab) and by following rules of division of a monomial by a monomial, we get the result as (\(2a^2\) – 3ab + \(4b^2\)).