Question 1189239
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Given that {{{ x^2-3x+2 }}} is a factor of {{{ x^4 + kx^3 - 10x^2 - 20x+24 }}} evaluate 
the sum of the four roots of the equation {{{ x^4 + kx^3 - 10x^2 - 20x+24 = 0 }}} 
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<pre>
Notice that x^2-3x+2 = (x-2)*(x-1).


We are given that the polynomial  x^4 + kx^3 - 10x^2 - 20x+24  is divisible by the polynomial  x^2-3x+2 .


Hence, the polynomial  x^4 + kx^3 - 10x^2 - 20x+24  is divisible by  (x-1).


It means (the Remainder theorem) that the number x= 1 is the root of the polynomial  x^4 + kx^3 - 10x^2 - 20x+24.


So, we substitute x= 1 into this polynomial, and we get this equation for "k"

    1^4 + k*1^3 - 10*1^2 - 20*1 + 24 = 0,

or

    1 + k - 10 - 20 + 24 = 0,

    k = 5.


Now use the Vieta's theorem: the sum of the roots of the polynomial  x^4 + kx^3 - 10x^2 - 20x + 24  is equal 
to the coefficient at x^3 with the opposite sign.


It gives that the sum of the roots of the polynomial  x^4 + kx^3 - 10x^2 - 20x+24   is equal to -k, i.e. -5.    <U>ANSWER</U>
</pre>

Solved.


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