Question

If alpha and beta are the zeroes of the quadratic polynomial f(x)=ax^2+bx+c then evaluate 1/alpha^3 + 1/beta^3
plz ans Its very urgent ​

Answer 1

Step-by-step explanation:

I will use m and n for alpha and beta

from the equation we know

m+n = -b/a

mn = c/a

(m+n)^2 = m^2 + n^2 + 2mn

so m^2 + n^2 = (m+n)^2 - 2mn

also

(m^2 + n^2)^2 = m^4 + n^4 + 2m^2n^2

so m^4 + n^4 = (m^2 + n^2)^2 - 2(mn)^2

=[(m+n)^2 - 2mn]^2 - 2(mn)^2

= [(-b/a)^2 - 2c/a]^2 - 2c^2/a^2

= [b^2/a^2 - 2c/a]^2 - 2c^2/a^2

let's test it with x^2 - 7x + 12 = 0

roots m and n , where m = 4, n=3

m+n= 7

mn = 12

according to my answer

m^4 + n^4 = [(49/1-2(12)]^2 - 2(144))/1 = 25^2 - 288 = 337

actual m^4 +n^4 = 4^4 + 3^4 = 256 + 81 = 337