Download Algebraic Fractions

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C3 Chapter 1: Algebraic Fractions
SIMPLIFYING AND THE FOUR OPERATIONS
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Factorise every term as far as possible.
To multiply fractions, cancel down any factors, and then multiply the numerators
and the denominators.
To divide, change into a multiplication by taking the reciprocal of the divisor.
We can only add/subtract fractions when they have the same denominator, so
change every term so that it has the same lowest common denominator
Where there are fractions in the numerator/denominator multiply throughout by
the lowest common denominator to eliminate these ‘fractions within fractions’.
EXAMPLE
a)
x2  6 x  9
12 x 2  42 x  18

3( x  2)(2 x  1)
4 x 2  36
b)
1
2

2x 1 x  3
c)
4( x  7) 2( x  7)(3x  2)

9( x 2  25)
15 x( x  5)
d)
3x  1
2
x
5
e)
6x
7x  2

(5  2 x)(6  x) (5  2 x)(1  x)
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REMAINDER THEOREM
An improper fraction (top heavy) is one whose numerator has a degree equal to or
greater than the denominator.
These can be changed into mixed numbers, either by long division or by using the
remainder theorem.
Remainder Theorem:
Any polynomial F(x) can be put in the form
F(x) = Q(x) × divisor + R(x)
where Q(x) is called the quotient function and R(x) is the remainder function.
Note that the degree of Q(x) can be found by subtracting the degree of the divisor
from the degree of F(x).
The degree of the remainder is always one less than the degree of the divisor.
EXAMPLE
Using the remainder theorem express the following as a mixed number.
x 4  2 x3  2 x 2  2 x  4
x2 1
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EXAMPLE
Given that ( x  2) is a factor of 2 x3  6 x 2  bx  5 find the remainder when the
expression is divided by (2 x  1) .