if alpha and beta are the zeroes of the polynomial 3x2+8x+2 find the value of, alpha3 +beta3 , alpha2-beta2 please answer this and i will mark you as brainliest
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if alpha and beta are the zeroes of the polynomial 3x2+8x+2 find the value of, alpha3 +beta3 , alpha2-beta2 please answer this and i will mark you as brainliest
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[tex] \large\bf\underline{Given:-}[/tex]
[tex] \large\bf\underline {To \: find:-}[/tex]
[tex] \huge\bf\underline{Solution:-}[/tex]
p(x) = 3x² + 8x + 2
≫ Sum of zeroes = -b/a
↣α + β = -8/3
≫ Product of zeroes = c/a
↣αβ = 2/3
Now, Finding value of α² + β²
we know that,
↣(a + b)² = a² + b² + 2ab
↣ (a + b)² - 2ab = a² + b²
↣a² + b² = (a + b)² - 2ab
≫ α² + β² = ( α + β)² - 2αβ....(i)
Substituting value of α + β and αβ in (i)
↣α² + β² = (-8/3)² - 2 × 2/3
↣α² + β² = 64/9 - 4/3
↣α² + β² = (64 - 12)/9
↣ α² + β² = 52/9
Now finding value of α³ + β³
we know that,
↣a³ + b³ = (a + b)(a² + b² - ab)
↣α³ + β³ = (α+ β) (α²+ β²-αβ)
Substituting value of α+ β , α²+ β² and αβ
↣α³ + β³ = (-8/3) × (52/9 - 2/3)
↣α³ + β³ = (-8/3) × (52-6)/9
↣α³ + β³ = (-8/3) × (46/9)
↣α³ + β³ = -368/27
hence
[tex]\rule{200}3[/tex]
[tex]\red{\color{white}{\fcolorbox{cyan}{black}{Answer:-}}}[/tex]
p(x) = 3x² + 8x + 2
a = 3, b = 8 & c = 2
Sum of zeroes = -b/a
[tex]\therefore[/tex] α + β = -8/3
Product of zeroes = c/a
[tex]\therefore[/tex] αβ = 2/3
1. α² + β²
We know that,
(a + b)² = a² + b² + 2ab
(a + b)² - 2ab = a² + b²
a² + b² = (a + b)² - 2ab
α² + β² = ( α + β)² - 2αβ⠀⠀⠀⠀⠀⠀(i)
Substituting value of α + β and αβ in (i) :-
α² + β² = (-8/3)² - 2 × 2/3
α² + β² = 64/9 - 4/3
α² + β² = (64 - 12)/9
[tex]\therefore[/tex] α² + β² = 52/9
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
2. α³ + β³
We know that,
a³ + b³ = (a + b)(a² + b² - ab)
α³ + β³ = (α+ β) (α²+ β²-αβ)
Putting value of α+ β , α²+ β² and αβ
α³ + β³ = (-8/3) × (52/9 - 2/3)
α³ + β³ = (-8/3) × (52-6)/9
α³ + β³ = (-8/3) × (46/9)
[tex]\therefore[/tex] α³ + β³ = -368/27