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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 + \(4b^2\)).