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Now, as per the division algorithm,
Divisor x Quotient + Remainder = Dividend
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(x – 3x + x + 2) = g(x) (x – 2) + (–2x + 4)
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(x – 3x + x + 2 + 2x -4) = g(x) (x – 2)
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(x – 3x + 3x - 2) = g(x) (x – 2)
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Hence, g(x) is the quotient when we divide (x – 3x + 3x – 2) by (x – 2).
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Therefore, g(x) = (x – x + 1).
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Example: Divide f(x) by g(x) and find the quotient and remainder when f(x)= 2x
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+ x + 3x + 2x + 20 and g(x) = x + 2x + 2. Also, state if g(x) is a factor of f(x) or
not.
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Given, f(x) = x + 2x + 3x + 2x + 20 and g(x) = x + 2x + 2
f(x) g(x)

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Now, quotient q(x) = x + 1 and r(x) = 18.
Since r(x) 0, therefore g(x) is not a factor of f(x).

*Factor theorem:- If p(x) is a polynomial of deg≥1 and p(a)=0 then (x – a) is a factor
of p(x).

*Remainder Theorem: If p(x) is a polynomial of deg≥1 and p(x) is divided by (x–a)
then p(a) is the remainder.

Method of finding the Remaining zeroes of a Polynomial When some of its
Zeroes are given:

Suppose a polynomial of degree 3 or 4 or more and its some zeroes, say two or
three are given, then to find other zeroes,

*we first write the factors of polynomial using the given zeroes and multiply them to
get g(x).

*Divide the given polynomial by g(x).

*The quotient so obtained gives other zeroes of given polynomial and we factorise it
(if possible) to get other zeroes.

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Example: Find the other two zeroes of polynomial x – 2x – 5x + 6 if x = 1 is
one of its zero.