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 polynomialIt is given that P= 1/p , Q=1/qThen,$$\displaystyle P+Q=\frac{1}{p}+\frac{1}{q}=\frac{q+p}{pq}=\frac{-b/a}{c/a}=\frac{-b}{c}$$$$\displaystyle PQ=\frac{1}{p}.\frac{1}{q}=\frac{1}{pq}=\frac{a}{c}$$$$\therefore$$ Required polynomial = $$\displaystyle x^{2}+\frac{b}{c}x+\frac{a}{c}\Rightarrow cx^{2}+bx+a$$Mathematics

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