580. To transform an equation into another whose roots are those of the original equation with their signs changed.
Let f(x) = 0 be the equation.
Put — y for x then the equation f(— y)= is satisfied by every root of f(x) = 0 with its sign changed ; tfus the required equation is f(— y)= 0.
If the given equation is then it is evident that the required equation will be therefore the transformed equation is obtained from the original equation by changing the sign of every alternate term beginning with the second.
Note. If any term of the given equation is missing it must be supplied with zero as a coefficient.
Ex. Transform the equation cc*-17x2-20x-6 = into another which shall have the same roots numerically with contrary signs. We may write the equation thus :
x4 + 0x3- 17a:2-20a:-6 = 0.
By the rule, we have
a;4 _ x3 - 17 x2 + 20 x - 6 = 0, or x4- 17x'^ + 20x-6 = 0.
581. To transform an equation into another whose roots are equal to those of the original equation multiplied by a given factor.
Let f(x) = 0 be the equation, and let q denote the given quantity. Put y = qx, so that when x has any particular value, y is q times as large; then x = {y}{q}, and the required equation is
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Multiplying by fy", we have
