1.Algebra Booster

2.6 Algebra Booster
3. To determine the nature of the roots we should remember
the following rules:
(a) If the co-efficients are all positive, the equation
has no positive root. Thus, the equation x 5 + 4x 3 +
2x + 1 = 0 has no positive root.
(b) If the co-efficients of the even powers of x are all
of the same sign and the co-efficients of the odd
powers of x are all of opposite sign, the equation
has no negative root.
For example, the equation x 7 + x 5 – 3x 4 + x 3 – 3x 2 +
2x – 5 = 0 has no negative root.
(c) If the equation contains only even powers of x and
the co-efficients are all of the same sign, the equation
has no real root. For example, the equation x 5
+ 3x 4 + 2x 2 + 1 = 0 cannot have a real root.
(d) If the equation contains only odd powers of x, and
the co-efficients are all of the same sign, the equation
has no real roots other than x = 0. For example,
the equation x 9 + 4x 5 + 5x 3 + 3x = 0 has no real
root other than x = 0.
1.19 DESCARTES RULE OF SIGNS
Consider a polynomial
n n-1 n-2
f( x) = a0x + a1x + a2x + ... + a n
with real co-efficients and a n
π 0, n Œ W.
We have
n n-1 n-2
f(–x) = a0( - x) + a1( - x) + a2( - x) + ..... + a n
By observing the sign variation, in the co-efficients of f(x)
and f(–x), we can predict the following things about the nature
of the roots.
(i) The maximum number of positive roots of f(x) is equal
to the number of sign changes in the co-efficients of
f(x). Let this number be p.
(ii) The maximum number of negative roots of f(x) is equal
to the number of sign changes in the co-efficients of
f(–x). Let this number be q.
(iii) The maximum number of imaginary roots of
f(x) = n – (p + q).
IN-EQUATIONS
2.1 CONCEPT OF SOLVING ALGEBRAIC IN-EQUATION
Solving an algebraic in-equation f(x) > 0, where f(x) is a product
(or quotient or both) of distinct linear factors. We proceed
methodically
(i) Make sure that the coefficients of x in all the linear factors
are positive.
(ii) Equate all factors to zero to get the critical points (the
points where these factors change their signs) c 1
, c 2
, c 3
,
… where c 1
< c 2
< c 3
< …
(iii) Write + and – sign alternately on the number line starting
from the extreme right.
(iv) If f(x) > 0, we shall use the open interval with the critical
points.
(v) If f(x) ≥ 0 or £ 0, we shall use the closed interval with
the critical points.
(vi) If f(x) contains an even power factors, say x 2 , (x – 2) 2 ,
(x – 3) 10 , (x – 4) 100 , we shall not represent them on the
number line, we neglect them.
(vii) If f(x) contains an irreducible quadratic factor (ax 2 + bx
+ c), where a > 0 and D < 0, we discard it.
2.2 EQUATION CONTAINING ABSOLUTE VALUES
Definition
Let for every x in R, the magnitude of x is called its modulus
or absolute value. It is denoted by |x|.
Ï x, x≥
0
Thus , f() x = || x = Ì .
Ó - x, x < 0
X¢
y = x
Properties of Modulus
O
Y
Y¢
y=
x
(i) Geometrically, the modulus of x means, it is the distance
between x and the origin, O.
(ii) ||x|| = |–x| = |x|
(iii) |x| = max (–x, x)
(iv) –|x| = min (–x, x)
(v)
x
2
= || x
(vi) |x + y| = |x| + |y|, when x and y are of the opposite sign.
fi xy £ 0
(vii) |x + y| = |x| + |y|, when x and y are of the same sign.
fi xy ≥ 0
(viii) |xy| = |x| |y|.
x | x| (ix) , y 0.
y = |y|
π
(x) |x| = 1 fi x = ±1
(xi) |x| = –2 fi x = j
(xii) |x| < 1 fi –1 < x < 1
(xiii) |x| > 1 fi x > 1 and x < –1
X