Question

if alpha and beta are the zeroes of the polynomial p(x) =x2-7x+12 then find the value of 1/alpha +1/beta​

Answer 1

Given :

If alpha and beta are the zeroes of the polynomial p(x) =x²-7x+12

To find :

$\sf \dfrac{1}{\alpha}+\dfrac{1}{\beta}$

Solution:

$\begin{gathered}\\ \quad\bullet\:{\sf{\purple{Given\: polynomial=x^2-7x+12}}} \\\end{gathered}$

$\begin{gathered}\\ \qquad\quad\large\mathfrak{\underline{As\:we\:know\:that}}\\\\\end{gathered}$

Quadratic Polynomial : ax² + bx + c, where a ≠ 0

$\begin{gathered}\\\\{\large\purple\dagger}\:{\bf{\pink{Sum\:of\:roots=\dfrac{-(coefficient\:of\:x)}{(coefficient\:of\:x^2)}}}}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \alpha+\beta=\dfrac{(-b)}{a}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \alpha+\beta=\dfrac{-(-7)}{1}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \alpha +\beta=7 \\\end{gathered}$

_______________________________

$\begin{gathered}\\{\large\purple\dagger}\:{\bf{\red{Product\:of\:roots=\dfrac{(constant\:term)}{(coefficient\:of\:x^2)}}}}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \alpha\times \beta=\dfrac{c}{a}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \alpha\times \beta=\dfrac{12}{1}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \alpha \beta=12 \\\end{gathered}$

_______________________________

Value of 1/a1/β

$\begin{gathered}\\\implies\sf \dfrac{1}{\alpha}+\dfrac{1}{\beta}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \dfrac{\beta+\alpha}{\alpha \beta}\\\\\end{gathered}$

$\begin{gathered}\implies\sf \dfrac{\alpha+\beta}{\alpha \beta}\\\end{gathered}$

• Substitute the valueso fα+β &αβ

$\begin{gathered}\\\implies\sf \dfrac{7}{12}\\\\\end{gathered}$

$\begin{gathered}\therefore{\underline{\boxed{\bf{Value\:of\: \dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{7}{12}}}}}\\\end{gathered}$

Answer 2

❥Answer᭄

α and β  are  zeroes of  polynomial  x² - 7 x + 12

     αβ =  12      the product of the roots  is the  constant term

   α + β = 7      the sum of the roots is the  coefficient of x term , negated.

1/α + 1/β + 2 αβ

     =  (α + β) / αβ  +  2 αβ

     =  12 / 7 + 2 *  12  =  24 + 12/7  =  180/7

Thank you