The equation `a_0 x^n + a_1 x^(n-1) + a_2x^(n-2) + ...... + a_(n-1)x + a_n = 0,` where
`a_0 , a_1 , a_2 , ......... , a_(n-1) , a_n` are constants but `a_0 ne 0 , ` is polynomial equation of
degree n . It has n only n roots . Let `alpha_1 , alpha_2 , alpha_3 ,............,alpha_(n-1) , alpha_n` be n roots , then
`sumalpha_1 = alpha_1 ,alpha_2 ,alpha_3 + ......... + alpha_(n-1) ,alpha_n = (-1)^1 a_1/a_0` [sum pf all roots]
`sumalpha_1 alpha_2= alpha_1 alpha_2+alpha_2 alpha_3 + ........ +alpha_1 alpha_n + alpha_2 alpha_3 + ......... + alpha_2 alpha_n + ........... + alpha_(n-1) alpha_1n `
`=(-1)^2a_2/a_0` [sum of products taken two at a time]
`sum alpha_1 alpha_2 alpha_3 = (-1)a_3/a_0` [sum of products taken three at a time]
and `alpha_1 alpha_2 alpha_3 ................. alpha_n = (-1)a_n/a_0` [product of all roots]
`text(In general)`
`sum alpha_1 alpha_2 alpha_3 ............. alpha_p = (-1)a_p/a_0`
`text(Important Result)`
1. A polynomial equation of degree n has n roots (real or imaginary).
2. If all the coefficients, i.e., `a_0 , a_1 , a_2, . , , a_n` are real, then the imaginary roots
occur in pairs, i.e. number of imaginary roots is always even.
3. If the degree of a polynomial equation is odd, then atleast one of the roots will
be real.
4. `(x-alpha_1)(x-alpha_2)(x-alpha_3).................(x-alpha_n)`
`= x^n + (-1)^1 sumalpha_1 . x^(n-1) + (-1)^2 sum alpha_1 alpha_2 . x^(n-2) + ...... + (-1) alpha_1 alpha_2 alpha_3 ............ alpha_n`
`text(For Cubic Equation, n=3 : )`
If `alpha, beta` and `gamma` are roots of a cubic equation `ax^3 + bx^2 + cx + d = 0` then
`ax^3 +bx^2 + cx + d = a(x - alpha)(x - beta)(x - gamma)`
`ax^3 + bx^2 + cx + d = a [x^3 - (sumalpha) x^2 + (sumalpha beta)x - alpha betagamma]`
Comparing co-efficients on both sides, we get
`alpha+ beta + gamma =-b/a`
`alpha beta + betagamma + gammaalpha = c/a`
and `\ \ \ \ \ \ \ \ \ \ \ \alpha betagamma =- d/a`
`text(For Bi-quadratic Equation , n=4:)`
If `alpha, beta` , `gamma` and `delta` are roots of a bi-quadratic equation `ax^4 + bx^3 + cx^2 + dx + e = 0` then
`ax^4 + bx^3 + cx^2 + dx + e = a(x - alpha) (x - beta )(x - gamma) (x - delta)`
`ax^4 + bx^3 + cx^2 + dx + e = a [x^4 - (sum alpha) x^3 + (sum alphabeta)x^2 - alphabeta gamma )x + alphabetagammadelta]`
Comparing co-efficients on both sides, we get
`alpha+ beta + gamma +delta=-b/a`
`alphabeta+alphagamma+alphadelta+beta gamma+beta delta+gammadelta=c/a`
`alphabetagamma+beta gammadelta+gammadeltaalpha+deltaalphabeta=-d/a`
and `\ \ \ \ \ \ \ \ \ \ \ \alpha betagamma delta=e/a`
The equation `a_0 x^n + a_1 x^(n-1) + a_2x^(n-2) + ...... + a_(n-1)x + a_n = 0,` where
`a_0 , a_1 , a_2 , ......... , a_(n-1) , a_n` are constants but `a_0 ne 0 , ` is polynomial equation of
degree n . It has n only n roots . Let `alpha_1 , alpha_2 , alpha_3 ,............,alpha_(n-1) , alpha_n` be n roots , then
`sumalpha_1 = alpha_1 ,alpha_2 ,alpha_3 + ......... + alpha_(n-1) ,alpha_n = (-1)^1 a_1/a_0` [sum pf all roots]
`sumalpha_1 alpha_2= alpha_1 alpha_2+alpha_2 alpha_3 + ........ +alpha_1 alpha_n + alpha_2 alpha_3 + ......... + alpha_2 alpha_n + ........... + alpha_(n-1) alpha_1n `
`=(-1)^2a_2/a_0` [sum of products taken two at a time]
`sum alpha_1 alpha_2 alpha_3 = (-1)a_3/a_0` [sum of products taken three at a time]
and `alpha_1 alpha_2 alpha_3 ................. alpha_n = (-1)a_n/a_0` [product of all roots]
`text(In general)`
`sum alpha_1 alpha_2 alpha_3 ............. alpha_p = (-1)a_p/a_0`
`text(Important Result)`
1. A polynomial equation of degree n has n roots (real or imaginary).
2. If all the coefficients, i.e., `a_0 , a_1 , a_2, . , , a_n` are real, then the imaginary roots
occur in pairs, i.e. number of imaginary roots is always even.
3. If the degree of a polynomial equation is odd, then atleast one of the roots will
be real.
4. `(x-alpha_1)(x-alpha_2)(x-alpha_3).................(x-alpha_n)`
`= x^n + (-1)^1 sumalpha_1 . x^(n-1) + (-1)^2 sum alpha_1 alpha_2 . x^(n-2) + ...... + (-1) alpha_1 alpha_2 alpha_3 ............ alpha_n`
`text(For Cubic Equation, n=3 : )`
If `alpha, beta` and `gamma` are roots of a cubic equation `ax^3 + bx^2 + cx + d = 0` then
`ax^3 +bx^2 + cx + d = a(x - alpha)(x - beta)(x - gamma)`
`ax^3 + bx^2 + cx + d = a [x^3 - (sumalpha) x^2 + (sumalpha beta)x - alpha betagamma]`
Comparing co-efficients on both sides, we get
`alpha+ beta + gamma =-b/a`
`alpha beta + betagamma + gammaalpha = c/a`
and `\ \ \ \ \ \ \ \ \ \ \ \alpha betagamma =- d/a`
`text(For Bi-quadratic Equation , n=4:)`
If `alpha, beta` , `gamma` and `delta` are roots of a bi-quadratic equation `ax^4 + bx^3 + cx^2 + dx + e = 0` then
`ax^4 + bx^3 + cx^2 + dx + e = a(x - alpha) (x - beta )(x - gamma) (x - delta)`
`ax^4 + bx^3 + cx^2 + dx + e = a [x^4 - (sum alpha) x^3 + (sum alphabeta)x^2 - alphabeta gamma )x + alphabetagammadelta]`
Comparing co-efficients on both sides, we get
`alpha+ beta + gamma +delta=-b/a`
`alphabeta+alphagamma+alphadelta+beta gamma+beta delta+gammadelta=c/a`
`alphabetagamma+beta gammadelta+gammadeltaalpha+deltaalphabeta=-d/a`
and `\ \ \ \ \ \ \ \ \ \ \ \alpha betagamma delta=e/a`