Publish On: 2019-04-10

Andy More

Total Post: 752

Question: Describe the common roots

Reply On: 2013-10-03

Mjay Jollay

Total Post: 0

ANS: Describe the common roots

Let ƒ(x) = 0 be an equation of degree n and Ø(x) = 0 be another equation of degree m. Let if possible

𝛼1, 𝛼2, ….., 𝛼D, p ≤ min. (m, n)

be the roots to both the equations ƒ(x) = 0 and Ø(x) = 0, then

ƒ(x) (x - 𝛼1)(x - 𝛼2) …. (x - 𝛼D) ƒD(x),             

where ƒD(x) is a polynomial of degree (n – p)

and Ø(x) = (x - 𝛼1)(x - 𝛼2) …. (x - 𝛼D) ØD(x),   

where ØD(x) is a polynomial of degree (m – p).

Also ƒD(x) and ØD(x) have no other common factor.

As (x - 𝛼1)(x - 𝛼2) …. (x - 𝛼D) is a factor common to both ƒD(x) and ØD(x) so it will divide both of them exactly and hence

(x - 𝛼1)(x - 𝛼2) …. (x - 𝛼D)

is the H.C.F. of ƒD(x) and ØD(x).

Therefore, in order to find the common roots of the two given equations we should find their H.C.F.

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