Theory of Equations..

GENERAL EQUATION OF N^th DEGREE

Let polynomial f(x) = a₀xⁿ + a₁x^{n – 1} + a₂x^{n – 2} + … + a_n. where a₀, a₁, a₂, ..a_n 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 n^th 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…

• 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.

• 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.

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