Question

if alpha and beta are the zeros of the polynomial X square + 7 X + 12 then find the value of 1/ alpha + 1/ beta - 2 alpha beta.​

Answer 1

Answer:

$\large\boxed{\sf{-\dfrac{295}{12}}}$

Step-by-step explanation:

Given polynomial,

${x}^{2} + 7x + 12$

Also, it's zeroes are $\alpha$ and $\beta$.

Therefore, we have, sum of zeroes,

$= > \alpha + \beta = - 7$

And, product of zeroes,

$\alpha \beta = 12$

Now, we have to find the value of,

$\dfrac{1}{ \alpha } + \dfrac{1}{ \beta } - 2 \alpha \beta$

Further simplifying, we will get,

$= \dfrac{ \alpha + \beta }{ \alpha \beta } - 2 \alpha \beta \\ \\ = \frac{ \alpha + \beta - 2 {( \alpha \beta )}^{2} }{ \alpha \beta }$

Substituting the respective values, we get,

$=\dfrac{ - 7 - 2 {(12)}^{2} }{12} \\ \\ = \dfrac{ - 7 - 2(144)}{12} \\ \\ = \dfrac{ - 7 - 288}{12} \\ \\ = - \dfrac{295}{12}$

Hence, the required value is $\bold{-\dfrac{295}{12}}$

Answer 2

$\huge\mathfrak\green{Answer:-}$

Given polynomial:

x² + 7 x + 12

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

=  -7/12 - 2 * 12 

=(-7/12) - 24/1

Taking LCM now,

→ (-7 - 24*12) / 12

→ (-7 - 288) / 12

→ (-295) / 12 (Ans).