Question

X=√2+√3+√6 is a root of x⁴+ax³+bx²+cx+d=0 then find the value of |a+b+c+d|

Answer 1

$\textbf{Given:}$

$\text{ x=\sqrt{2}+\sqrt{3}+\sqrt{6} is a root of x^4+ax^3+bx^2+cx+d }$

$\textbf{To find:}$

$\text{The value of}\;|a+b+c+d|$

$\textbf{Solution:}$

$\text{Consider,}$

$x=\sqrt{2}+\sqrt{3}+\sqrt{6}$

$x-\sqrt{2}=\sqrt{3}+\sqrt{6}$

$\text{Squaring on bothsides, we get}$

$(x-\sqrt{2})^2=(\sqrt{3}+\sqrt{6})^2$

$x^2+2-2\sqrt{2}\,x=3+6+2\,\sqrt{3}\,\sqrt{6}$

$x^2+2-2\sqrt{2}\,x=9+2\,\sqrt{3}\,\sqrt{6}$

$\text{Rearranging terms, we get}$

$x^2-7=2\sqrt{2}\,x+2\,\sqrt{3}\,\sqrt{6}$

$\text{Squaring again on bothsides, we get}$

$(x^2-7)^2=(2\sqrt{2}\,x+2\,\sqrt{3}\,\sqrt{6})^2$

$x^4+49-14\,x^2=8x^2+72+48x$

$\text{Rearranging terms, we get}$

$x^4-22\,x^2-48x-23=0$

$\text{Comparing it with x^4+ax^3+bx^2+cx+d we get}$

$a=0,\;b=-22,\;c=-48,\;d=-23$

$\implies\,a+b+c+d=-93$

$\implies\,|a+b+c+d|=|-93|$

$\implies\,|a+b+c+d|=93$

$\therefore\textbf{The value of \bf|a+b+c+d| is 93}$

Find more:

If a,b,c are rational then the root of equation (a+2b-3c )x²+(b+2c-3a)x+ (c+2a-3b)=0

https://brainly.in/question/4813981

Given that a=alpha and b=beta are roots of 2x^2+x+14=0 find the equation whose roots are a^4 and b^4

https://brainly.in/question/17088379

If two of the roots of equation x4 - 2x3 + ax2 + 8x + b = 0 are equal in magnitude but opposite in sig

then value of 4a + b is equal to :

(A) 16

(B) 8

(C) -16

theme values are also

28 ** ** *5 = o are equal in magnitude but opposite in

(D) -8​

https://brainly.in/question/10136947

Answer 2

mod(a+b+c+d) is 93

Step-by-step explanation:

its a PYQ of PRMO