Math, asked by josephbovas0404, 3 months ago

If x⁵-5x⁴ + 9x³ + ax² + bx + c = 0 is a reciproca
equation find a, b, c.​

Answers

Answered by farhaanaarif84
1

Answer:

Given equation, x

5

−5x

4

+9x

3

−9x

2

+5x−1=0

Consider f(x) = x^5-5x^4+9x^3-9x^2+5x-1

Notice that this is a reciprocal equation of odd degree which has the opposite signs of the first and last term.

∴(x−1) is one factor of the given equation and the quotient is another reciprocal function which has same signs of the first and last term.

∴f(x)=(x−1)(Ax

4

+Bx

3

+Cx

2

+Bx+A)

Comparing the coefficient, we have A=1,B=−4,C=5

⟹f(x)=(x−1)(x

4

−4x

3

+5x

2

−4x+1)

Consider g(x)=x

4

−5x

3

−22x

2

−5x+1=(x

4

+1)−4(x

3

+x)+5x

2

We need to find the roots of g(x)=0

⟹(x

2

+x

−2

)–4(x+x

−1

)+5=0[dividing byx

2

]

Substitute x+x

−1

=y in the above equation

⟹(y

2

−2)−4y+5=0⟹y

2

−4y+3=0⟹(y−3)(y−1)=0

∴x+x

−1

=3 and x+x

−1

=1

Solving the first quadratic equations we have, x=

2

5

Solving the second quadratic equations we have, x=

2

3

i

∴ roots of the given equation are 1,

2

5

,

2

3

i

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