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THEORY OF EQUATIONS

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(1) For a cubic equation ax^3+bx^2+cx+d=o

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

(2) For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0

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

(3) 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.

(4) Besides Complex roots , even irrational roots occur in pairs. Hence if 2+root(3) is a root , then even 2-root(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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