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Question

Find the quadratic polynomial whose zeros are reciprocals of the zeros of the polynomials $$f(x)=ax^{2}+bx
+c, \ a\neq 0, \ c\neq 0$$.


Solution

Let p,q be zeros of $$ax^{2}+bx+c$$

$$\therefore p+q=\dfrac{-b}{a}$$  &  $$pq=\dfrac{c}{a}$$

Let P & Q be zeros of required polynomial

It is given that P= 1/p , Q=1/q

Then,
$$\displaystyle P+Q=\frac{1}{p}+\frac{1}{q}=\frac{q+p}{pq}=\frac{-b/a}{c/a}=\frac{-b}{c}$$

$$\displaystyle PQ=\frac{1}{p}.\frac{1}{q}=\frac{1}{pq}=\frac{a}{c}$$

$$\therefore $$ Required polynomial = $$\displaystyle x^{2}+\frac{b}{c}x+\frac{a}{c}\Rightarrow cx^{2}+bx+a$$

Mathematics

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