Sunday, September 28, 2008

Theory of Equations..


Let polynomial f(x) = a0xn + a1xn – 1 + a2xn – 2 + … + an. where a0, a1, a2, are rational numbers and n ³ 0. Then the values of x for which f(x) reduces to zero are called root of the equation f(x) = 0. The highest whole number power of x is called the degree of the equation.

For example

x4 – 3x3 + 4x2 + x + 1 = 0 is an equation with degree four.

x5 – 6x4 + 3x2 + 1 = 0 is an equation with degree five.

ax + b = 0 is called the linear equation.

ax2 + bx + c = 0 is called the quadratic equation.

ax3 + bx2 + cx + d = 0 is called the cubic equation.

Properties of equations and their roots

  • Every equation of the nth degree has exactly n roots.

For example, the equation x3 + 4x2 + 1 = 0 has 3 roots,

The equation x5 – x + 2 = 0 has 5 roots, and so on…

nature of roots

  • In an equation with real coefficients imaginary roots occur in pairs i.e. if a + ib is a root of the equation f(x) = 0, then a – ib will also be a root of the same equation. For example, if 2 + 3i is a root of equation f(x) = 0, 2 – 3i is also a root.

surd roots occur in pairs

  • If the coefficients of an equation are all positive then the equation has no positive root. Hence, the equation 2x4 + 3x2 + 5x + 1 = 0 has no positive root.
  • If the coefficients of even powers of x are all of one sign, and the coefficients of the odd powers are all of opposite sign, then the equation has no negative root. Hence, the equation 6x4 – 11x3 + 5x2 – 2x + 1 = 0 has no negative root
  • If the equation contains only even powers of x and the coefficients are all of the same sign, the equation has no real root. Hence, the equation 4x4 + 5x2 + 2 = 0 has no real root.
  • If the equation contains only odd powers of x, and the coefficients are all of the same sign, the equation has no real root except x = 0. Hence, the equation 5x5 + 4x3 + x = 0 has only one real root at x = 0.
  • Descartes’ Rule of Signs : An equation f(x) = 0 cannot have more positive roots than there are changes of sign in f(x), and cannot have more negative roots than there changes of sign in f( - x). Thus the equation x4 + 7x3 − 4x2 − x – 7 = 0 has one positive root because there is only change in sign. f( - x) = x4 − 7x3 − 4x2 + x – 7 = 0 hence the number of negative real roots will be either 1 or 3.



theory of equations problems

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