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SAI International School
Class-X
Mathematics
Chapter-2: Polynomials

Lesson Notes

Sub topics: Division Algorithm for Polynomials

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) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

Note: Degree p(x) = Degree g(x) + Degree q(x)
Degree of a constant Polynomial is 0.

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Example: Divide 3x – x – 3x + 5 by x – 1 – x , and
verify the division algorithm.

Solution :
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On dividing 3x – x – 3x + 5 by x – 1 – x , we get,
Terms of Dividend and divisor should be arranged in Descending order before
division.

Here, quotient is (x – 2) and remainder is 3.
Now, as per the division algorithm, Divisor x Quotient + Remainder = Dividend
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LHS = (-x + x + 1)(x – 2) + 3
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= (–x + x – x + 2x – 2x + 2 + 3)
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= (–x + 3x – 3x + 5)
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RHS = (–x + 3x – 3x + 5)
Thus, division algorithm is verified.
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Example: On dividing x – 3x + x + 2 by a polynomial g(x), the quotient and
remainder were (x – 2) and (–2x + 4), respectively. Find g(x).
Sol:
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Given, dividend = p(x) = (x – 3x + x + 2), quotient = (x – 2), remainder = (–2x + 4).
Let divisor be denoted by g(x).