Page 1 - Lesson Note-3 Reamainder and Factor Theorem Ch-2 POLYNOMIAL

SAI International School
CLASS - IX

Mathematics
CHAPTER-2: POLYNOMIALS -3 Lesson Notes-3

Sub Topic:

1. Division of polynomial 2. Remainder Theorem 3. factor theorem

1. We know the property of division which follows in the basic division, i.e.

Dividend = (Divisor × Quotient) + Remainder
This same follows the division of polynomial.

If p(x) and g(x) are two polynomials in which the degree of p(x) ≥ degree of g(x) and
g(x) ≠ 0 are given then we can get the 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).

It says that p(x) divided by g(x), gives q(x) as quotient and r(x) as remainder.
2.Remainder Theorem:
If p(x) be any polynomial of degree greater than or equal to one and let ‘a’ be any
real number and p (x) is divided by the linear polynomial x – a, then the remainder is
p(a).

As we know that

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

If p(x) is divided by (x-a) then,

If x = a
P (a) = (a -a). q (a) + r = 0

To find the remainder or to check the multiple of the polynomial we can use the
remainder theorem.

Example:
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4
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What is the remainder if a + a – 2a + a + 1 is divided by (a – 1).
Solution:
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3
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P(x) = a + a – 2a + a + 1
To find the zero of the (a – 1) we need to equate it to zero.
a −1 = 0
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