Let f(x) be a (non-constant) polynomial over a number system K, then a complex number α is called a zero (or root) of f(x) iff f(α) = 0.
A zero of the polynomial f(x) is also called a root of the pilynomial or algebraic equation f(x) = 0.
Common Roots: To find common roots of two algebraic equations f(x) =0, g(x) = 0.
Let f(x) = 0 and g(x) = 0 be all algebraic equations of degrees m and n respectively.
Let α1, α2, ... , αr (r ≤ m,r ≤ n) be all the common roots of the two given equations, then both f(x) and g(x) are divisble by (x - α1)(x - α2) ... (x - αr) which is, therefore, the H.C.F of f(x) and g(x).
Hence, the common roots of the two equations f(x) = 0 and g(x) = 0 are given by d(x) = 0 where d(x) is the H.C.F. of f(x) and g(x).
Multiplicity of a Root: Let f(x) = 0 be an algebraic equation of degree n (≥ 1), then α is called a root of multiplicity m(≤ n) of f(x) = 0 iff there exists a polynomial g(x) such that
f(x) = (x – α)mg(x), g(α) ≠ 0
In particular, m = 0 iff α is not a root of f(x) = 0, and α is called a simple root of f(x) = 0 iff m=1, and α is called a repeated root of f(x) = 0 iff m > .
If m = 2, α is called a double root and if m = 3, α is called a triple root etc.
Relations Between The Roots and The Coefficients:
Let α1, α2, ... , αn be any number, then
(x- α1)(x- α2) = x2 - (α1 + α2)x + α1α2 = x2 - σ1x + σ2
where σ1 = α1 + α2 = ∑α1 = sum of number αi's taken one at a time
and σ2 = α1α2
(x- α1)(x- α2)(x- α3) = x3 - (α1 + α2 + α3)x2 + (α1α2 + α1α3 + α2α3)x - α1α2α3
= x3 - σ1x2 + σ2x - σ3
where σ1 = α1 + α2 + α3 = ∑α1 = sum of number αi's taken one at a time,
σ2 = α1α2 + α1α3 + α2α3 = ∑α1α2
= sum of products of numbers αi's taken two at a time
σ3 = α1α2α3 and so on.
Generalising,
(x- α1)(x- α2)(x- α3) ... (x- αn) = xn - σ1xn-1 + σ2xn-2 + ... + (-1)rσrxn-r + ... + (-1)nσn
where σ1 = α1 + α2 +... + αn = ∑α1 = sum of number αi's taken one at a time,
σ2 = α1α2 + α1α3 + ... + α1αn + α2α3 + ... + α2αn + αn-1αn = ∑α1α2
= sum of products of numbers αi's taken two at a time
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
σr = sum of products of numbers αi's taken r at a time
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
σn = α1α2α3 ... αn.
A zero of the polynomial f(x) is also called a root of the pilynomial or algebraic equation f(x) = 0.
Common Roots: To find common roots of two algebraic equations f(x) =0, g(x) = 0.
Let f(x) = 0 and g(x) = 0 be all algebraic equations of degrees m and n respectively.
Let α1, α2, ... , αr (r ≤ m,r ≤ n) be all the common roots of the two given equations, then both f(x) and g(x) are divisble by (x - α1)(x - α2) ... (x - αr) which is, therefore, the H.C.F of f(x) and g(x).
Hence, the common roots of the two equations f(x) = 0 and g(x) = 0 are given by d(x) = 0 where d(x) is the H.C.F. of f(x) and g(x).
Multiplicity of a Root: Let f(x) = 0 be an algebraic equation of degree n (≥ 1), then α is called a root of multiplicity m(≤ n) of f(x) = 0 iff there exists a polynomial g(x) such that
f(x) = (x – α)mg(x), g(α) ≠ 0
In particular, m = 0 iff α is not a root of f(x) = 0, and α is called a simple root of f(x) = 0 iff m=1, and α is called a repeated root of f(x) = 0 iff m > .
If m = 2, α is called a double root and if m = 3, α is called a triple root etc.
Relations Between The Roots and The Coefficients:
Let α1, α2, ... , αn be any number, then
(x- α1)(x- α2) = x2 - (α1 + α2)x + α1α2 = x2 - σ1x + σ2
where σ1 = α1 + α2 = ∑α1 = sum of number αi's taken one at a time
and σ2 = α1α2
(x- α1)(x- α2)(x- α3) = x3 - (α1 + α2 + α3)x2 + (α1α2 + α1α3 + α2α3)x - α1α2α3
= x3 - σ1x2 + σ2x - σ3
where σ1 = α1 + α2 + α3 = ∑α1 = sum of number αi's taken one at a time,
σ2 = α1α2 + α1α3 + α2α3 = ∑α1α2
= sum of products of numbers αi's taken two at a time
σ3 = α1α2α3 and so on.
Generalising,
(x- α1)(x- α2)(x- α3) ... (x- αn) = xn - σ1xn-1 + σ2xn-2 + ... + (-1)rσrxn-r + ... + (-1)nσn
where σ1 = α1 + α2 +... + αn = ∑α1 = sum of number αi's taken one at a time,
σ2 = α1α2 + α1α3 + ... + α1αn + α2α3 + ... + α2αn + αn-1αn = ∑α1α2
= sum of products of numbers αi's taken two at a time
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
σr = sum of products of numbers αi's taken r at a time
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
σn = α1α2α3 ... αn.
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