Write the product 2 below 5 and add. The sum is 7. Multiply 7 with 2 and write the
product 14 below 6. The sum is 20. Hence in this case the remainder is 20 and the
quotient is x + 7.
- Remainder theorem
When a polynomial P(x) of degree greater than 1, is divided by a binomial (x-a) where
‘a’ is a real number, then the remainder is P(a).
Eg. :
Let P(x) = x^2 -5x+6
Dividing P(x) by x-1
Then P(1) = 12 – 5(1)+6
= 1-5+6
= 7-5
P(1) = 2
i.e., the remainder when x^2 -5x+6 is divided by x-1 is 2.
Unlike in the synthetic division where both the quotient and the remainder can be
found, in order to find the remainder alone the remainder theorem may be used.
In the above example the polynominal P(x) = x^2 –5x –16, which has to be divided by
x –1. Consider x –1 = 0 ⇒ x = 1. Hence in P(x), x has to be replaced by 1, thus
finding P(1). Replacing x by 1 and simplifying leads to the value P(1) = 2. Thus the
remainder when x^2 – 5x + 6 is divided by x–1 is 2.
- Factor theorem
When a polynominal P(x) of degree greater than 1 is divided by a binominal (x-a), and
if P(a)=0 then (x-a) is a factor of a polynominal P(x).
Eg. :
Let P(x) = x^2 +5x+6
