Question

find the zeros of the polynomial f(x)=x³- 5x²-2x+24 what it's at given product of two zeros is 12 , -2

Answer 1

Let α,β and y be the zeros of polynomial f(x) such that ab=12

We have, α+β+y=

a

−b

=

1

−(−5)

=5

αβ+βy+yα=

a

c

=

1

−2

=−2 and αβy=

a

−d

=

1

−24

=−24

Putting αβ=12 in αβy=−24, we get

12y=−24 ⇒ y=−

12

24

=−2

Now ,α+β+y=5 ⇒ α+β−2=5

⇒ α+β=7 ⇒ α=7−β

∴αβ=12

⇒(7−β)β=12 ⇒

7β−β

2

=12

⇒β

2

−7β+12=0 ⇒ β

2

−3β−4β+12=0

⇒β=4 or β=3

∴α=3 or α=4