(x^4 + 2 + 7x^2 + x^3 - 9x^5) ÷ (x+4)
Answers
Step-by-step explanation:
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
3±
5
Solving the second quadratic equations we have, x=
2
1±
3
i
∴ roots of the given equation are 1,
2
3±
5
,
2
1±
3
i