Question

if alpha and beeta are the zeroes of the polynomial x2+4x+3,from the polynomial whose zeroes are 1+beeta/alphaand 1+alpha/beeta

Answer 1

Given polynomial is x²+4x+3
Sum of the roots, α + β = -b/a = -4
Product of the roots, αβ = c/a = 3

Now,

Sum of roots of $\frac{1+ \beta }{ \alpha }, \frac{1+ \alpha }{ \beta }$

$\frac{1+ \beta }{ \alpha } + \frac{1+ \alpha }{ \beta }$

$\frac{ \beta + \beta ^2+ \alpha + \alpha ^2}{ \alpha \beta }$

$\frac{ \alpha ^2+ \beta ^2+ \alpha + \beta }{ \alpha \beta }$

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

$\frac{(-4)^2-2(3)+(-4)}{3}$

$\frac{16 - 6 - 4}{3}$

= 2

Product of the roots of $\frac{1+ \beta }{ \alpha } , \frac{1+ \alpha }{ \beta }$

$\frac{1+ \alpha \beta + \alpha + \beta }{ \alpha \beta }$

$\frac{1+3-4}{3}$

= 0

The polynomial is x² - (sum of the roots)x + (Product of the roots)

⇒ x² - 2x + 0

⇒ x² - 2x