Shortcuts to getting information about the roots

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1
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If an equation contains all positive coefficients of any powers of x, then it has no
positive roots.[/i
[
e.g.
has no positive roots
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2
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If all the even powers of x have same sign coefficients and all the odd powers of x
have the opposite sign coefficients, then the equation has no negative roots
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e.g
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Summarizing DESCARTES rules of signs
:
For an equation f(x)=0, the maximum number of positive roots it can have is the
number of sign changes in f(x); and the maximum number of negative roots it can
have is the number of sign changes in f(x
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4
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Consider the two equations
ax + by = c
dx + ey = f
Then, If , then we have infinite solutions for these equations
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If
, then we have no solution for these equations
.
If
, then we have a unique solutions for these equations
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Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i ,
another has to be 23i and if there are three possible roots of the equation, we can
conclude that the last root is real. This real roots could be found out by finding the
sum of the roots of the equation and subtracting (2+3i)+(23i)=4 from that sum
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If an equation f(x)= 0 has only odd powers of x and all these have the same sign
coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign
coefficients then the equation has no real roots in each case, except for x=0 in the
second case
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7
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Besides Complex roots, even irrational roots occur in pairs. Hence if 2+root(3) is a
root, then even 2root(3) is a root . (All these are very useful in finding number of
positive, negative, real, complex etc roots of an equation
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8
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x + y >= x+y ( stands for absolute value or modulus ) (Useful in solving some
inequations
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For a cubic equation
sum of the roots =  b/a
sum of the product of the roots taken two at a time = c/a
product of the roots = d/a
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10
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For a biquadratic equation
sum of the roots =  b/a
sum of the product of the roots taken three at a time = c/a
sum of the product of the roots taken two at a time = d/a
product of the roots = e/a
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View Full DocumentGeometry
General Notions and useful shortcuts
:
Polygons
:
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1
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For any regular polygon, the sum of the interior angles is equal to 360 degrees
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2
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If any parallelogram can be inscribed in a circle , it must be a rectangle
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2.1
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Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram, the coordinates of
the meeting point of the diagonals can be found out by solving for [(a+e)/2, (b+f)/2]
=[ (c+g)/2, (d+h)/2
[
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3
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If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e
oblique sies equal
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4
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For an isosceles trapezium , sum of a pair of opposite sides is equal in length to
the sum of the other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order
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 Spring '10
 RazaSuleman
 Addition, triangle, Quadrilateral

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