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

1.5.1 Properandimproperfractions
1.5 Algebraic fractions 27
Whenpresentedwithafraction,wecannotethedegreeofthenumerator,sayn,andthe
degree of the denominator, sayd.
A fraction is proper ifd > n, that is the degree of the denominator is greater than
the degree of the numerator. Ifd nthen the fraction isimproper.
Example1.25 Classify the following fractions as either proper or improper. In each case, state the
degree of both numerator and denominator.
x 2 +9x−6
(a)
3x 3 +x 2 +100
(b) t3 +t 2 +9t−6
t 5 +9
(v +1)(v −6)
(c)
v 2 +3v+6
(z +2) 3
(d)
5z 2 +10z +16
Solution (a) The degree of the numerator,n, is 2. The degree of the denominator,d, is 3. Since
d >nthe fraction isproper.
(b) Heren = 3 andd = 5.The fraction isproper sinced >n.
(c) Heren = 2 andd = 2,sod =nand the fraction isimproper.
(d) Heren = 3 andd = 2,sod <nand the fraction isimproper.
1.5.2 Equivalentfractions
Consider the numerical fractions 1 2 and 2 . These fractions have the same value. Similarly,
2 3 , 6 20
4
and
9 30 all have the same value. The algebraic fractions x y , 2x xt
and
2y yt all
have the same value. Fractions with the same value arecalledequivalent fractions.
The value of a fraction remains unchanged if both numerator and denominator are
multiplied or divided by the same quantity. This fact can be used to write a fraction in
many equivalent forms.Consider forexample the fractions
(a) 2 x
(b)
2(x +1)
x(x +1)
(c) 2xt
x 2 t
These are all equivalent fractions. Fraction (b) can be obtained by multiplying both numerator
and denominator of fraction (a) by (x +1), so they are equivalent. Fraction (a)
canbeobtainedbydividingnumeratoranddenominatoroffraction(c)byxt andsothey
arealsoequivalent.
Example1.26 Show that
x +1
x +7
areequivalent.
and
x 2 +4x+3
x 2 +10x+21