Name: Introduction to Polynomial Functions
Uploaded: 2019-07-19
Duration: 4 min 57 s
Description: The roots of a polynomial are defined as the points at which the polynomial becomes zero. Can we use the roots in graphing polynomial functions? Watch this v...

Question

If

a, b, c

are the roots of

x^{4} + p x^{3} + q x^{2} + r x + s = 0,

find the value of

Σ a^{4}

.

Accepted Answer

Given, $x^{4} + p x^{3} + q x^{2} + r x + s = 0$

Therefore, $Σ a = - p, Σ a b = q, Σ a b c = - r, a b c d = s$

We have to calculate $Σ a^{4}$

$∵ Σ a^{2} Σ a^{2} = Σ a^{4} + 2 Σ a^{2} b^{2}$

$∴ Σ a^{4} = {(\sum a^{2})}^{2} - 2 Σ a^{2} b^{2}$

$= {[{(\sum a)}^{2} - 2 Σ a b]}^{2} - 2 Σ a^{2} b^{2} . . . (i)$

$∴ a^{2} = (Σ a)^{2} - 2 Σ a b$

Again $(Σ a b)^{2} = Σ a^{2} b^{2} + 2 Σ a^{2} b c + 6 a b c d$

$∴ Σ a^{2} b^{2} = (Σ a b)^{2} - 2 Σ a^{2} b c - 6 a b c d$

$= q^{2} - 2 Σ a^{2} b c - 6 s . . . (i i)$

Also, $Σ a Σ a b c = Σ a^{2} b c + 4 a b c d$

So, $Σ a^{2} b c = (- p) (- r) - 4 s$ $= p r - 4 s$

$∴$ $Σ a^{2} b^{2} = q^{2} - 2 p r + 8 s - 6 s$ $= q^{2} - 2 p r + 2 s$

Hence expression $(i)$ becomes

$Σ a^{4} = {(p^{2} - 2 q)}^{2} - 2 q^{2} = 4 p r - 4 s$

$= p^{4} - 4 p^{2} q + 4 q^{2} - 2 q^{2} + 4 p r - 4 s$

$= p^{4} - 4 p^{2} q + 2 q^{2} + 4 p r - 4 s$ .

If $a, b, c$ are the roots of $x^{4} + p x^{3} + q x^{2} + r x + s = 0,$ find the value of $Σ a^{4}$ .

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

If a,b,c are the roots of x4+px3+qx2+rx+s=0, find the value of Σa4.

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

If $a, b, c$ are the roots of $x^{4} + p x^{3} + q x^{2} + r x + s = 0,$ find the value of $Σ a^{4}$ .