Factor Completely: x^4-6x^3-27x^2​

8x

4

−2x

3

−27x

2

+6x+9=0

Let the roots be a,−a,b,c

S

1

=a−a+b+c=−

8

−2

=

4

1

b+c=

4

1

(i)

S

3

=−a

2

b−abc+abc−a

2

c=−

8

6

⇒a

2

(b+c)=

4

3

(ii)

dividing (i) by (ii)

⇒b+c=

4

1

(iii)

S

4

=−a

2

bc=

8

9

⇒bc=−

8

3

(iv)

substituting b from (iii) in (i)

b(

4

1

−b)=−

8

3

4

b−4b

2

=−

8

3

2b−8b

2

=3

8b

2

−2b−3=0

8b

2

+4b−6b−3=0

4b(2b+1)−3(2b+1)=0

(4b−3)(2b+1)=0

⇒b=

4

3

,−

2

1

substituing b in (iv)

⇒c=−

2

1

,

4

3

S

2

=a(−a)+(−a)(b)+bc+ca=

8

−27

⇒−a

2

−ab+bc+ac=

8

−27

The value of s

4

is not satisfied for b=−

2

1

So the roots of the equation are

3

,−

3

,

4

3

,−

2

1

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