Question

Find the zeroes of the polynomial f(x)=x^3- 5x^2-2x + 24, If it is given that the product of its two zeroes is 12.​

Answer 1

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

we have α + β + y = -b/a = -(-5)/1 = 5

αβ + βy + yα = -c/a = -2/1 = -2 and,

αβy = -d/a = -24/1 = -24

putting αβ = 12 in αβy = -24

we get,

=> 12y = -24

=> y = -24/12

=> y = -2

Now, α + β + y = 5

=> α + β + (-2) = 5

=> α + β = 7

=> α = 7 - β

since, αβ = 12

=> (7 - β)β = 12

=> 7β - β² = 12

=> β² - 7β - 12 = 0

=> β² - 3β - 4β - 12 = 0

=> β = 4 or β = 3

Therefore, α = 4 or α = 3

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Answer 2

Answer:

Step-by-step explanation:

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

 ab = 12

we have α + β + y = -b/a 

= -(-5)/1 

= 5αβ + βy + yα = -c/a 

= -2/1 

= -2 

and,αβy = -d/a 

= -24/1 

= -24

putting αβ = 12 in αβy = -24

we get,=> 12y = -24=> y = -24/12=> y = -2

Now, α + β + y = 5=> α + β + (-2) = 5=> α + β = 7=> α = 7 - β

since, αβ = 12=> (7 - β)

β = 12

=> 7β - β² = 12

=> β² - 7β - 12 = 0

=> β² - 3β - 4β - 12 = 0

=> β = 4 or β = 3

Therefore, α = 4 or α = 3