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

EXPLANATION.

α and β are the zeroes of the polynomial,

⇒ p(x) = x² - 7x + 12 = 0.

As we know that,

Sum of the zeroes of the quadratic equation,

⇒ α + β = -b/a.

⇒ α + β = -(-7)/1 = 7. ⇒ (1).

Products of the zeroes of the quadratic equation,

⇒ αβ = c/a.

⇒ αβ = 12/1 = 12.

Value of 1/α + 1/β.

⇒ 1/α + 1/β.

Take L.C.M in equation, we get.

⇒ β + α/αβ.

Put the value in equation, we get.

⇒ 7/12.

                                                                                       

MORE INFORMATION.

Conjugate roots,

If D < 0,

one roots = α + iβ.

other roots = α - iβ.

If D > 0,

one roots = α + √β.

other roots = α - √β.

Answer 2

α and β are zeroes of polynomial ,

x² - 7x + 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.

$\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \: \: \frac{1}{ \alpha} + \frac{ 1}{ \beta} + 2 \alpha \beta$

$\\ \\ \sf \: \: \: \: \: \: \: \: \: \: : \implies \: \bigg( \: \frac{ \alpha \ + \beta}{ \alpha \beta \: } \bigg)+ 2 \alpha \beta$

→ 12 / 7 + 2× 12 = 24 + 12 /7 = 180