Find all rational zeros of f. Then (if necessary) use the depressed equation to find all roots of the equation f(x)=0 x^4-2x^3-43x^2-82x-24=0

Nola Aguilar

Nola Aguilar

Answered question

2022-11-13

Find all rational zeros of f. Then (if necessary) use the depressed equation to find all roots of the equation f(x)=0
x 4 2 x 3 43 x 2 82 x 24 = 0

Answer & Explanation

grizintimbp

grizintimbp

Beginner2022-11-14Added 16 answers

Given polynomial is f ( x ) = x 4 2 x 3 43 x 2 82 x 24 = 0
Here the leading coefficient is 1.
Then the possible rational zeros are the interger factors of theconstant term -24
They are ± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 8 , ± 12 , ± 24
Then we have x 4 2 x 3 43 x 2 82 x 24 = ( x + 2 ) ( x 3 4 x 2 35 x 12 )
The depressed polynomial is x 3 4 x 2 35 x 12
Use the quadratic formula x = b ± b 2 4 a c 2 a for x 2 8 x 3:
here a=1,b=-8, c=-3
x = ( 8 ) ± ( 8 ) 2 4 ( 3 ) 2 = 8 ± 76 2 = 8 ± 2 19 2 = 4 ± 19
Then we have
x 4 2 x 3 43 x 2 82 x 24 = ( x + 2 ) ( x 3 4 x 2 35 x 12 ) = ( x + 2 ) ( x + 4 ) ( x 2 8 x 3 )
So, x = 2 , 4 , 4 ± 19

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?