find the zeros of the polynomial f of x is equal to X raised to the power of 3 - 5 x raised ^ 2 - 2 X + 24, if it is given that the product of two zeros is 12
Answers
Answer :-
3, 4 and - 2 are the zeroes
Solution :-
f( x ) = x³ - 5x² - 2x + 24
Comparing with ax³ + bx² + cx + d we get ,
- a = 1
- b = - 5
- c = - 2
- d = 24
Let α, β and γ be zeroes of the polynomial
Given : Product of two zeroes = 12
⇒ αβ = 12
⇒ β = 12/α
Product of zeroes = αβγ = - d/a = - 24/1 = - 24
⇒ 12 * γ = - 24
⇒ γ = - 24/12 = - 2
Sum of zeroes = α + β + γ = - b/a = - ( - 5 )/ 1 = 5
⇒ α + β + ( - 2 ) = 5
⇒ α + β = 5 + 2 = 7
⇒ α + β = 7
⇒ α + 12/α = 7
⇒ α² + 12 = 7α
⇒ α² - 7a + 12 = 0
⇒ α² - 3α - 4α + 12 = 0
⇒ α(α - 3) - 4(α - 3) = 0
⇒ (α - 4)(α - 3) = 0
⇒ α - 4 = 0 or α - 3 = 0
⇒ α = 4 or α = 3
When α = 4
⇒ β = 7 - 4 = 3
When α = 3
⇒ β = 7 - 3 = 4
Therefore the zeroes of the polynomial are 4, 3 and - 2.
Answer:
Step-by-step explanation:
Given :-
f(x) = x³ - 5x² - 2x + 24
Product of two zeroes = 12
To Find :-
Zeros of Polynomial.
Solution :-
f(x) = x³ - 5x² - 2x + 24
Comparing with ax³ + bx² + cx + d,
a = 1
b = - 5
c = - 2
d = 24
Let α, β and γ be zeroes of the polynomial
⇒ αβ = 12
⇒ β = 12/α
Product of zeroes = αβγ = - d/a = - 24/1 = - 24
⇒ 12 × γ = - 24
⇒ γ = - 24/12 = - 2
Sum of zeroes = α + β + γ = - b/a = - ( - 5 )/ 1 = 5
⇒ α + β + ( - 2 ) = 5
⇒ α + β = 5 + 2 = 7
⇒ α + β = 7
⇒ α + 12/α = 7
⇒ α² + 12 = 7α
⇒ α² - 7a + 12 = 0
⇒ α² - 3α - 4α + 12 = 0
⇒ α(α - 3) - 4(α - 3) = 0
⇒ (α - 4)(α - 3) = 0
⇒ α - 4 = 0 or α - 3 = 0
⇒ α = 4 or α = 3
When α = 4
⇒ β = 7 - 4
⇒ β = 3
When α = 3
⇒ β = 7 - 3
⇒ β = 4
Hence, the zeroes of the polynomial are 4, 3 and -2.