Page 3 - 10-Math-4 PARTIAL FRACTIONS

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if degree of the polynomial N(x) is greater or equal to the degree of Version 1.1
the polynomial D(x).

e.g., are improper fractions.

Every improper fraction can be reduced by division to the sum of a

polynomial and a proper fraction. This means that if degree of the

numerator is greater or equal to the degree of the denominator,

then we can divide N(x) by D(x) obtaining a quotient polynomial Q(x)

and a remainder polynomial R(x), whose degree is less than the

degree of D(x).

Thus , with D(x) ≠0. Where Q(x) is quotient

polynomial and is a proper fraction. For example, x2 +1 is an
x +1
improper fraction.
x2 +1 = 1 i.e., an improper fraction x2 +1 has been
∴ x +1 (x +1) + x +1
x +1
resolved to a quotient polynomial Q(x) = x − 1 and a proper

fraction

Example 1: Resolve the fraction into proper fraction.

Solution: Let N(x) = x3 − x2 + x + 1 and D(x) = x2 + 5
x −1

By long division, we have x2 + 5 x3 − x2 + x + 1
−x3 ± 5x

− x2 − 4x +1

 x2  5

− 4x + 6

Activity: Separate proper and improper fractions

(i) x 2+ x + 1 (ii) 2x + 5 (iii) x3 + x2 + 1 (iv) 2x
x2 + 1 +1)(x + x3 −1 1)( x
( x 2) ( x − − 2)

Activity: Convert the following improper fractions into

proper fractions.

(i) 3x2 − 2x −1 (ii) 6x3 +5 x2 − 6
2 2x 2− x −1

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