082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

1.5 Algebraic fractions 29
(b) Factorizing and cancelling common factors yields
6t 3 +3t 2 +6t
3t 2 +3t
= 3t(2t2 +t+2)
3t(t +1)
= 2t2 +t+2
t +1
Note thatthe common factor, 3t, has been cancelled.
4t+8
Example1.29 Simplify(a) (b) 2y2 −y−1
t 2 +3t+2 y 2 −2y+1
Solution The numerator and denominator arefactorized and common factors are cancelled.
4t+8
(a)
t 2 +3t+2 = 4(t +2)
(t +2)(t +1) = 4
t +1
(b)
The common factor,t +2,has been cancelled.
2y 2 −y−1 (2y +1)(y −1)
= = 2y+1
y 2 −2y+1 (y −1) 2 y −1
The common factor,y−1,has been cancelled.
1.5.4 Multiplicationanddivisionofalgebraicfractions
Tomultiplytwoalgebraicfractionstogether,wemultiplytheirnumeratorstogether,and
multiplytheirdenominatorstogether, thatis
a
b × c d = a ×c
b ×d = ac
bd
Division isperformed by inverting the second fraction and then multiplying, thatis
a
b ÷ c d = a b × d c = ad
bc
Before multiplying or dividing fractions it is advisable to express each fraction in its
simplest form.
Example1.30 Simplify
x 2 +5x+6
2x−2
× x2 −x
x 2 +3x+2
Solution Factorizing numerators and denominators produces
x 2 +5x+6
2x−2
× x2 −x (x +2)(x +3) x(x −1)
= ×
x 2 +3x+2 2(x −1) (x +1)(x +2)
=
(x +2)(x +3)x(x −1)
2(x −1)(x +1)(x +2)
Commonfactors (x +2)and (x −1)canbecancelledfromnumeratoranddenominator
togive
(x +2)(x +3)x(x −1) (x +3)x
=
2(x −1)(x +1)(x +2) 2(x +1)