Find all real zeros of the polynomial. use quadratic formula. P ( x ) =...

Darian Hubbard

Darian Hubbard

Answered

2022-07-28

Find all real zeros of the polynomial. use quadratic formula. P ( x ) = x 5 8 x 4 + 7 x 3 + 30 x 2 + 2 x 12

Answer & Explanation

Franklin Frey

Franklin Frey

Expert

2022-07-29Added 15 answers

Consider the polynomial P ( x ) = x 5 8 x 4 + 7 x 3 + 30 x 2 + 2 x 12
Factor the polynomial as
P ( x ) = x 3 ( x 2 8 x 2 + 7 ) + 30 x 2 + 2 x 12 = ( x + 1 ) 2 ( x 6 ) ( x 2 4 x + 2 )
Now to find real zeros, set P(x)=0 and solve for x.
P ( x ) = 0 ( x + 1 ) 2 ( x 6 ) ( x 2 4 x + 2 ) = 0 ( x + 1 ) 2 = 0 x = 1
Now solve the equation x 2 4 x + 2 = 0 using quadratic formula x = b ± b 2 4 a c 2 a as
x = ( 4 ) ± ( 4 ) 2 4 ( 1 ) ( 2 ) 2 ( 1 ) = 4 ± 8 2 = 4 ± 2 2 2 = 2 ± 2
Therefore, the real zeros are [ 1 , 6 , 2 2 , 2 + 2 ]

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