Question

if Alpha and beta are the zeroes of the quadratic polynomial p(x)=6x²-7x+12, find a quadratic polynomial whose zeroes are 1/aplha and 1/beta​

Answer 1

We are given that $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial p(x) = 6x² - 7x + 12

6x² - 7x + 12

$\longrightarrow$ a = 6, b = (- 7), c = 12

Sum of zeroes = $\sf{ \bf{ \dfrac{- b}{a}}}$

$\longrightarrow \: \sf{ \alpha + \beta = \dfrac{-( - 7)}{6}}$

$\longrightarrow \: \sf{ \alpha + \beta = \dfrac{ 7}{6}}$

Product of zeroes = $\sf{ \bf{ \dfrac{c}{a}}}$

$\longrightarrow \: \sf{ \alpha \times \beta = \dfrac{12}{6}}$

$\longrightarrow \: \sf{ \alpha \times \beta = 2}$

To Find :

• The quadratic polynomial whose zeroes are $\dfrac{1}{\alpha}$ and $\dfrac{1}{\beta}$

Sum of zeroes :

$\sf{\dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{ \beta + \alpha}{ \alpha \beta} = \dfrac{ \frac{7}{6} }{2} = \dfrac{7}{12}}$

Product of zeroes :

$\sf{\dfrac{1}{\alpha} \times \dfrac{1}{\beta} = \dfrac{ 1}{ \alpha \beta} = \dfrac{ 1}{2}}$

As we know : P(x) = x² - (sum of zeroes)x + (product of zeroes)

Required Polynomial = x² - $\dfrac{7}{12}$x + $\dfrac{1}{2}$