**GENERAL EQUATION OF N ^{th} DEGREE **

Let polynomial f(x) = a_{0}x^{n} + a_{1}x^{n – 1} + a_{2}x^{n – 2} + … + a_{n}. where a_{0}, a_{1}, a_{2}, ..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

x^{4} – 3x^{3} + 4x^{2} + x + 1 = 0 is an equation with degree four.

x^{5} – 6x^{4} + 3x^{2} + 1 = 0 is an equation with degree five.

ax + b = 0 is called the linear equation.

ax^{2} + bx + c = 0 is called the quadratic equation.

ax^{3} + bx^{2} + 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 x^{3} + 4x^{2} + 1 = 0 has 3 roots,

The equation x^{5} – 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 2x
^{4}+ 3x^{2}+ 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 6x
^{4}– 11x^{3}+ 5x^{2}– 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 4x^{4}+ 5x^{2}+ 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 5x^{5}+ 4x^{3}+ 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 x^{4}+ 7x^{3}− 4x^{2}− x – 7 = 0 has one positive root because there is only change in sign. f( - x) = x^{4}− 7x^{3}− 4x^{2}+ x – 7 = 0 hence the number of negative real roots will be either 1 or 3.

**EXAMPLES:**

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