Some Other Forms of Equations

RELATION BETWEEN THE ROOTS OF A POLYNOMIAL EQUATION OF DEGREE N

Consider the equation

a_nxⁿ + a_{n – 1}x^{n – 1} + a_{n – 2}x^{n – 2} + …. + a₁x + a₀ = 0 . . . . (1)

( where a₀, a₁…., a_n are real coefficients and a_n $\ne$ 0)

Let $\alpha$₁, $\alpha$₂,….,$\alpha$_n be the roots of equation (1). Then

a_nxⁿ + a_{n – 1}x^{n – 1} + a_{n – 2}x^{n – 2} + ….. + a₁x + a­₀ º a_n(x - $\alpha$₁) (x - $\alpha$₂) ….. (x - $\alpha$_n)

Comparing the coefficients of like powers of x, we get

$\alpha$₁ + $\alpha$₂ + $\alpha$₃ + …. + $\alpha$_n = -x$\frac{a_{n-1}}{a_n}$.

$\alpha$₁$\alpha$­₂ + $\alpha$₁$\alpha$₃ + $\alpha$₁$\alpha$₄ + …. + $\alpha$₂$\alpha$₃ + … + $\alpha$_{n - 1}$\alpha$_n = $\frac{a_{n-2}}{a_n}$

………………………………

$\alpha$₁$\alpha$­₂ . . . . .$\alpha$_r + …. + $\alpha$_n-r+1$\alpha$_n-r+2 … $\alpha$_n = ( -1)^{r$\frac{a_{n-r}}{a_n}$}

…….…………………………

$\alpha$₁$\alpha$₂ … $\alpha$_n = (-1)^{n$\frac{a_0}{a_n}$}

e.g. If $\alpha$, $\beta$, $\gamma$ and $\delta$ are the roots of ax⁴ + bx³ + cx² + dx + e = 0 then

$\alpha$ + $\beta$ + $\gamma$ + $\delta$ = -b/a

$\alpha$$\beta$ + $\alpha$$\gamma$+ $\alpha$$\delta$ + $\beta$$\gamma$ + $\beta$$\delta$ + $\gamma$$\delta$ = c/a

$\alpha$$\beta$$\gamma$ + $\alpha$$\beta$$\delta$ + $\alpha$$\gamma$$\delta$ + $\beta$$\gamma$$\delta$ = -d/a

$\alpha$$\beta$$\gamma$$\delta$ = e/a
Some important results:
$\bullet$ A polynomial equation of degree n has n roots (real or imaginary).
$\bullet$ If all the coefficients are real then the imaginary roots occur in pairs i.e. number of complex roots is always even.
$\bullet$ If the degree of a polynomial equation is odd then the number of real roots will also be odd. It follows that at least one of the roots will be real.
$\bullet$Factor theorem: If $\alpha$ is a root of the equation f(x)=0, then f(x) is exactly divisible by (x–$\alpha$) and conversely, if f(x) is exactly divisible by (x–$\alpha$) then $\alpha$ is a root of the equation f(x)=0.
$\bullet$ Let f(x)=0 be a polynomial equation and p and q are two real numbers, then f(x)=0 will have at least one real root or an odd number of roots between p and q if f(p) and f(q) are of opposite sign. But if f(p) and f(q) are of same signs, then either f(x)=0 has no real roots or an even number of roots between p and q.
$\bullet$ If $\alpha$ is repeated root repeating r times of a polynomial equation f(x) = 0 of degree n i.e. f(x) = (x - $\alpha$)^r g(x) , where g(x) is a polynomial of degree n - r and g($\alpha$)$\ne$0,then f($\alpha$) = f'($\alpha$) = f''($\alpha$) = . . . . = f ^(r-1)($\alpha$) = 0 and f ^r ($\alpha$) $\ne$ 0.
The cubic function f(x) = ax³ + bx² + cx + d, where x $\in$ R Take a > 0, the graph of f has the following properties:
$\bullet$As x $\to$ $\infty$, y $\to$ $\infty$ because the x³ term is positive and will dominate the remaining terms when x is large.
$\bullet$As x $\to$ $\infty$, y $\to$ –$\infty$
$\bullet$A consideration of (i) and (ii) implies that the graph of f must cross the x–axis at least once, taking this point x = $\beta$ (say), we have ax³ + bx² + cx + d = (x – $\beta$)Q where Q is a quadratic expression in x.
$\bullet$The equation Q = 0 may have two real distinct roots, two real coincident roots or no real roots.
$\bullet$f'(x) = 3ax² + 2bx + c and the equation f'(x) = 0 may have two real distinct roots, two real coincident roots or no real roots.
$\bullet$If f'(x) = 0 has two real distinct roots p and q (say) where p > q, f(p) is a minimum value of f(x) and f(q) is maximum value of f(x).
$\bullet$If f'(x) = 0 has two real coincident roots, r (say) then the stationary value of f(x) at (r, f(r)) is a point of inflexion.
$\bullet$If f'(x) = 0 has no real roots, the graph of f has no stationary points.f'(x) = 6ax + 2b and f''(x) = 0, f'''(x) $\ne$ 0 when x = –$\frac{b}{3a}$This implies that all cubic curves have a point of inflexion.
Illustration 1: If b² < 2ac, then prove that ax³ + bx² + cx + d=0 has exactly one real root.
Solution1: Let $\alpha$, $\beta$, $\gamma$ be the roots of ax³ +bx² +cx +d =0 Then $\alpha$ + $\beta$ + $\gamma$ = $-\frac{b}{a}$, $\alpha$$\beta$ + $\beta$$\gamma$ + $\gamma$$\alpha$ =$\frac{c}{a}$ and $\alpha$$\beta$$\gamma$ = $\frac{-d}{a}$

Also $\alpha$² +$\beta$² +$\gamma$² = ($\alpha$ + $\beta$ + $\gamma$ )² – 2($\alpha$$\beta$ + $\beta$$\gamma$ +$\gamma$$\alpha$) = $\frac{b^2}{a^2}-\frac{2c}{a}=\frac{b^2-2ac}{a^2}$.

$\Rightarrow$ $\alpha$² +$\beta$² + $\gamma$² < 0 , which is not possible if all $\alpha$, $\beta$,$\gamma$ are real.

So atleast one root is non-real, but complex roots occurs in pair. Hence given cubic equation has two non-real and one real roots.

Solution 2: Let f(x) = ax³ + bx² + cx + d and f(x) = 0 has all the roots real

$\Rightarrow$ f' (x) = 3ax² +2bx +c = 0 has two real roots

But its discriminant = (2b)² – 4. 3.ac = (b² – 2ac) – ac < 0 ( as b² < 2ac )

which is a contradiction$\Rightarrow$ f(x) =0 will not have all the roots real.