If α and β are zeroes of the polynomial P(x) = x² – 5x + 4, find the value of ³ + ³
and ³ − ³
Hint : use identities.
solve this for me its really important
Answers
Correct Question:
If α and β are zeroes of the polynomial P(x) = x² - 5x + 4, find the value of α³ + β³ and α³ - β³.
Answer:
The value of α³ + β³ is 65.
The value of α³ - β³ is ± 63.
Step-by-step explanation:
Given that:
α and β are zeroes of the polynomial P(x) = x² - 5x + 4.
To Find:
The value of α³ + β³.
The value of α³ - β³.
We have:
x² + (- 5)x + 4 is in the form of ax² + bx + c.
Where,
- a = 1
- b = - 5
- c = 4
We know that:
Sum of zeroes = α + β = - b/a = - (- 5)/1 = 5
Product of zeroes = αβ = c/a = 4/1 = 4
Finding the value of α³ + β³:
α³ + β³ = (α + β)³ - 3αβ(α + β)
α³ + β³ = (5)³ - 3•4(5)
α³ + β³ = 125 - 60
α³ + β³ = 65
∴ The value of α³ + β³ = 65
Finding the value α³ - β³:
α³ - β³ = (α - β) (α² + β² + αβ)
α³ - β³ = √{(α - β)²} {(α + β)² - 2αβ + αβ}
α³ - β³ = √{(α + β)² - 4αβ} {(α + β)² - 2αβ + αβ}
α³ - β³ = √{(5)² - 4•4} {(5)² - 4}
α³ - β³ = √{25 - 16} {25 - 4}
α³ - β³ = √{9} {21}
α³ - β³ = ± 3 × 21
α³ - β³ = ± 63
∴ The value of α³ - β³ = ± 63
Algebraic identities used:
- a³ + b³ = (a + b) - 3ab(a + b)
- a³ - b³ = (a - b) (a² + b² + ab)
- a² + b² = (a + b)² - 2ab
- (a - b)² = (a + b)² - 4ab
Step-by-step explanation:
Correct Question:
If α and β are zeroes of the polynomial P(x) = x² - 5x + 4, find the value of α³ + β³ and α³ - β³.
Answer:
The value of α³ + β³ is 65.
The value of α³ - β³ is ± 63.
Step-by-step explanation:
Given that:
α and β are zeroes of the polynomial P(x) = x² - 5x + 4.
To Find:
The value of α³ + β³.
The value of α³ - β³.
We have:
x² + (- 5)x + 4 is in the form of ax² + bx + c.
Where,
a = 1
b = - 5
c = 4
We know that:
Sum of zeroes = α + β = - b/a = - (- 5)/1 = 5
Product of zeroes = αβ = c/a = 4/1 = 4
Finding the value of α³ + β³:
α³ + β³ = (α + β)³ - 3αβ(α + β)
α³ + β³ = (5)³ - 3•4(5)
α³ + β³ = 125 - 60
α³ + β³ = 65
∴ The value of α³ + β³ = 65
Finding the value α³ - β³:
α³ - β³ = (α - β) (α² + β² + αβ)
α³ - β³ = √{(α - β)²} {(α + β)² - 2αβ + αβ}
α³ - β³ = √{(α + β)² - 4αβ} {(α + β)² - 2αβ + αβ}
α³ - β³ = √{(5)² - 4•4} {(5)² - 4}
α³ - β³ = √{25 - 16} {25 - 4}
α³ - β³ = √{9} {21}
α³ - β³ = ± 3 × 21
α³ - β³ = ± 63
∴ The value of α³ - β³ = ± 63
Algebraic identities used:
a³ + b³ = (a + b) - 3ab(a + b)
a³ - b³ = (a - b) (a² + b² + ab)
a² + b² = (a + b)² - 2ab