If α and β are the zeros of the polynomial f(x)=5x^2+4x−9 then evaluate the following: alpha^3+beta^3
Answers
Given equation
5x² + 4x - 9 = 0
To find the value of
α³ + β³
By comparing with
ax² - bx + c = 0
we get
a = 5 , b = 4 and c = -9
We know that
Sum of the zeroes = α + β = -b/a
Product of the zeroes = αβ = c/a
we have
α + β = -4/5
αβ = -(-9)/5 = 9/5
now we have to find
α³ + β³
We know that
(α + β)³ = α³ + β³ + 3αβ( α + β)
(α + β)³ - 3αβ( α + β) = α³ + β³
Put the value
(α + β)³ - 3αβ( α + β)
(-4/5)³ - 3×9/5(-4/5)
-64/125 - 27/5(-4/5)
-64/125 + 108/25
-64/125 + 108×5/125
-64/125 + 540/125
476/125
Answer
476/125
Given:-
- α and β are the zeros of the polynomial
To Find:-
- Value of α³ + β³
Formula used:-
- (a + b)² = a² + b² + 2ab
- (a - b)² = a² + b² - 2ab
- a³ + b³ = (a + b) (a² - ab + b²)
Solution:-
Comparing the given equation with standard equation i.e. ax² + bx + c = 0 we get,
- a = 5
- b = 4
- c = -9
Sum of zeroes =
Product of zeroes =
Using identity, (a + b)² = a² + b² + 2ab
Putting values,
Now, using identity a³ + b³ = (a + b) (a² - ab + b²)
Hence, The value of
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