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multiple correct type: if f(x)=x^4+x^3-2x^2-8x then which of the following are correct options :a- f(x)=0 has two real roots, b- f(x)=0has real roots in (-1,2),c- sum of real roots of f(x)=0 is 2 , d- product of real roots of f(x)=0 is 2.please sir explain to me each option as i am not able to.

multiple correct type: if f(x)=x^4+x^3-2x^2-8x then which of the following are correct options :a- f(x)=0 has two real roots, b- f(x)=0has real roots in (-1,2),c- sum of real roots of f(x)=0 is 2 , d- product of real roots of f(x)=0 is 2.please sir explain to me each option as i am not able to.

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1 Answers

Y RAJYALAKSHMI
45 Points
9 years ago
f(x) = x^4 + x^3 – 2x^2 – 8x
There is one change of sign in f(x), ie, +x^3 & -2x^2 – so we can have atmost 1 positive real root
f(-x) = x^4 - x^3 – 2x^2 + 8x
There are two one change of sign in f(-x), ie, x^4, –x^3 & -2x^2 ,+8x –  so we can have atmost 2 negative real roots.
 
But f(x) is of 4th degree.  So, we can have 4 roots.  Since the complex roots are always in pairs, we can have only two real root (either +ve or -ve)
 
x(x^ 3 + x^2 – 2x -8) = 0
=> x = 0 or x^ 3 + x^2 – 2x -8 = 0
by verifying we find the other root is 2 
So, the real roots are 0, 2 does not belong to (-1, 2) 
Sum of real roots 0+2 = 2.
product of real roots = 0 * 2 = 0
 
Ans:  a, c, d
 
 

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