Theory of Polynomial Equations

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Unit 9

• Theory of Polynomial Equations

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9.1 Relation between Roots and Coefficients of a
Polynomial Equation
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9.1 Relation between Roots and Coefficients of a
Polynomial Equation
Fundamental theorem of algebra
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9.1 Relation between Roots and Coefficients of a
Polynomial Equation
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9.1 Relation between Roots and Coefficients of a
Polynomial Equation
Particular cases
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9.1 Relation between Roots and Coefficients of a
Polynomial Equation
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9.1 Relation between Roots and Coefficients of a
Polynomial Equation
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P.292 Ex.9A
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9.2 The Sums of Powers of the Roots of an
Equation (extension)
Newtons formulas for the sums of powers of the
roots
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P.297 Ex.9B
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9.3 Multiple (Repeated) Roots
A polynomial equation f(x) 0 has a repeated
root ? if and only if f(?) f (?) 0, i.e. ?
is a common root root f(x) 0 and f (x) 0
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9.3 Multiple (Repeated) Roots
Taylors Formula
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9.3 Multiple (Repeated) Roots
Another form Taylors Formula
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9.3 Multiple (Repeated) Roots
? is a k-multiple root of the polynomial equation
f(x) 0 if and only if f(x) is divisible by (x -
?)k but not divisible by (x - ?)k1.
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9.3 Multiple (Repeated) Roots

• Let f(x) be a polynomial of degree n( ).
  Then
• Any single root of the equation f(x) 0 is not a
  root of the equation f (x) 0.

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9.3 Multiple (Repeated) Roots
Let f(x) and g(x) be two polynomials. (a) h(x) is
a common factor of f(x) and g(x) if and only if
both f(x) and g(x) are divisible by h(x). The
common factor with highest degree is the highest
common factor, abbreviated H.C.F. ( or Greatest
Common Divisor G.C.D.)
e.g. f(x) x(x 1)2(x - 1)3 g(x) x2(x
1)3(x - 2) HCF x(x 1)2
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9.3 Multiple (Repeated) Roots
(b) f(x) and g(x) are relatively prime if and
only if f(x) and g(x) have no common factor other
than a constant common factor.
e.g. x3 1 and x2 x 2 are relatively prime.
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9.3 Multiple (Repeated) Roots

• Let f(x) be a polynomial of degree .
• If f(x) and f (x) are relatively prime, then the
  equation f(x) 0 has only single roots.

(2) If the highest common factor of f(x) and
f (x) contains a k-multiple factor (x - ?),
then ? is a (k 1)-multiple root of the
equation f(x) 0.
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P.304 Ex.9C
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9.4 Rational and Irrational Roots

• Rational Roots

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9.4 Rational and Irrational Roots

• Rational Roots

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9.4 Rational and Irrational Roots

• Rational Roots

Let f(x) xn b1xn-1 bn-1x bn 0 be an
equation with integral coefficients, then the
rational roots of f(x) 0 can only be integers.
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9.4 Rational and Irrational Roots

• Rational Roots

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P.306 Ex.9D
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9.5 Sign of the Roots (extension)

• Let a0, a1,, an be positive numbers then
• the equation a0xn a1xn-1, an 0 has no
  positive root.
• the equation a0xn - a1xn-1, (-1)nan 0 has
  no negative root.

The sum of positive numbers and sum of negative
numbers cannot be zero.
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9.5 Sign of the Roots
Let ? be a positive number, and f(x) a
polynomial. If the coefficients of all the terms
of the quotient and the remainder are positive
when f(x) is divided by (x - ?), then the
equation f(x) 0 has no root greater than or
equal to ?.
e.g. Solve x4 4x3 7x2 10x 20 0
The equation has no positive integral root.
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9.6 Existence of Real Roots
Let f(x) be a polynomial with real coefficients.
If f(a) and f(b) are of different signs, where a
and b are real numbers with a lt b, then there are
odd number of real roots (including repeated
roots), lying between a and b, of the equation
f(x) 0.
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9.6 Existence of Real Roots
Let f(x) be a polynomial with real coefficients.
If f(a) and f(b) have the same sign where a and b
are real numbers with a lt b, then there are even
number of real roots (including repeated roots)
or no real root of the equation f(x) 0, lying
between a and b.
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P.311 Ex.9E
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9.7 Transformation of Equations
4 cases for changing the values of roots
(i) Roots with signs changed
If ? is a root of the polynomial equation f(x)
0, then -? must be a root of the equation f(-x)
0. That is , to transform the equation f(x) 0
into another whose roots are the same as those of
f(x) 0 but with opposite signs it is only
necessary to replace by x by x.
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9.7 Transformation of Equations
4 cases for changing the values of roots
(ii) Roots multiplied by a given non-zero
constant k.
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9.7 Transformation of Equations
4 cases for changing the values of roots
(iii) Roots increased by a given constant h.
If ? is a root of the polynomial equation f(x)
0, then ? h must be a root of the equation f(x
- h) 0.
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9.7 Transformation of Equations
4 cases for changing the values of roots
(iii) Roots increased by a given constant h.
The transformation may give rise to a new
equation with one or more missing terms.
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9.7 Transformation of Equations
4 cases for changing the values of roots
(iv) Reciprocal Roots.
If ? is a non-zero root of the polynomial
equation f(x) 0, then must be a root of
the equation f( ) 0.
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9.7 Transformation of Equations
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P.314 Ex.9F
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P.314 Ex.9G
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Ex.9G No.7
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Ex.9G No.7