Download Further Engineering Mathematics: Programs and Problems by K. STROUD PDF

By K. STROUD

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* a different technique-oriented strategy takes the scholar during the arithmetic in a hugely available way
* finished insurance of all issues required by means of undergraduates at complicated degrees of arithmetic in engineering and science
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F'(a) 4. Equation with given roots If rx, ß, y ... are roots, then the equation is (x-rx)(x-ß)(x-y) ... = 0 38 Further Engineering Mathematics 50 Relation between coefficients and roots If ~, ß, y are the roots of x 3 + px 2 +qx +r = 0 ~+ß+y=-p ~ß +ßy +y~ ~ßy In general, if ~' ß, y 0 0 0 = q = -r are roots of the equation PoX" +P1Xn-l +p2xn-l +o oo +Pn-1X +p" = 0 then sum of the roots = - Pt Po sum of products of roots, taken two at a time = - Pl Po sum of products of roots, taken three at a time = - PJ Po sum of products of roots, n at a time = (- 1)" Pn Po 60 Transformation of equations f(x) = ax 4 +bx 3 +cx 2 +dx +c = 0 lf the new roots are the old roots increased by k, ioeo y = x + k, substitute x = y- k and simplify f(y) = a(y-kt +b(y-k) 3 +c(y-k) 2 +d(y-k) +c = 0 etco 7 Reciprocal equations-unaltered when the variable is replaced by its reciprocal.

In Ill, the equationf(x) = 0 has one real root at A and two complex roots corresponding to B and C-these being a pair of conjugate complex roots. Whatever the form of the roots of a cubic equation, one root at least is always........... Programme 2: Theory of Equations 45 If the one real root, x = k, can be found,f(x) can be factorised into (x- k)(ax 2 + px +q) by long division, so that the equation f(x) = 0 becomes (x- k)(ax 2 +px +q) = 0. ·. x = k or ax 2 + px + q = 0 which can always be solved by factors or formula.

JO 3. Remainder theorem If f(x) = a 1 xn +a2xn-l + ... +anx +an+ I then f (k) = the remainder resulting from dividing f (x) by (x- k). If f(k) = 0, then (x- k) is a factor of f(x). e. f'(a) 4. Equation with given roots If rx, ß, y ... are roots, then the equation is (x-rx)(x-ß)(x-y) ... = 0 38 Further Engineering Mathematics 50 Relation between coefficients and roots If ~, ß, y are the roots of x 3 + px 2 +qx +r = 0 ~+ß+y=-p ~ß +ßy +y~ ~ßy In general, if ~' ß, y 0 0 0 = q = -r are roots of the equation PoX" +P1Xn-l +p2xn-l +o oo +Pn-1X +p" = 0 then sum of the roots = - Pt Po sum of products of roots, taken two at a time = - Pl Po sum of products of roots, taken three at a time = - PJ Po sum of products of roots, n at a time = (- 1)" Pn Po 60 Transformation of equations f(x) = ax 4 +bx 3 +cx 2 +dx +c = 0 lf the new roots are the old roots increased by k, ioeo y = x + k, substitute x = y- k and simplify f(y) = a(y-kt +b(y-k) 3 +c(y-k) 2 +d(y-k) +c = 0 etco 7 Reciprocal equations-unaltered when the variable is replaced by its reciprocal.

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