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

𝛼

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

ƒ(x) (x - 𝛼

where ƒ

and Ø(x) = (x - 𝛼

where Ø

Also ƒ

As (x - 𝛼

(x - 𝛼

is the H.C.F. of ƒ

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

𝛼

_{1}, 𝛼_{2}, ….., 𝛼_{D}, p ≤ min. (m, n)be the roots to both the equations ƒ(x) = 0 and Ø(x) = 0, then

ƒ(x) (x - 𝛼

_{1})(x -**𝛼**_{2}) …. (x - 𝛼_{D}) ƒ_{D}(x),where ƒ

_{D}(x) is a polynomial of degree (n – p)and Ø(x) = (x - 𝛼

_{1})(x - 𝛼_{2}) …. (x - 𝛼_{D}) Ø_{D}(x),where Ø

_{D}(x) is a polynomial of degree (m – p).Also ƒ

_{D}(x) and Ø_{D}(x) have no other common factor.As (x - 𝛼

_{1})(x - 𝛼_{2}) …. (x - 𝛼_{D}) is a factor common to both ƒ_{D}(x) and Ø_{D}(x) so it will divide both of them exactly and hence(x - 𝛼

_{1})(x - 𝛼_{2}) …. (x - 𝛼_{D})is the H.C.F. of ƒ

_{D}(x) and Ø_{D}(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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