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if alpha and beta are the zeroes of the polynomial p(x)=x²-5x+6,find the value of alpha⁴beta²+alpha²beta⁴
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Answers
Answered by
102
Given:-
→ Polynomial : P(x) = x² - 5x + 6
To find:-
- Value of α⁴β² + α²β⁴
Solution:-
Here, p(x) = 0
∴ x² - 5x + 6 = 0
→ x² - (3 + 2)x + 6 = 0
→ x² - 3x - 2x + 6 = 0
→ x(x - 3) - 2(x - 3) = 0
→ (x - 3)(x - 2) = 0
At first,
→ (x - 2) = 0
→ x = 2
Again,
→ (x - 3) = 0
→ x = 3
Therefore,
Zeros of polynomial are: 3 & 2
Here,
- α = 3
- β = 2
Now we have to find value of :
→ α⁴β² + α²β⁴
Putting values we get:-
→ (3)⁴ × (2)² + (3)² × (2)⁴
→ (81 × 4) + (9 × 16)
→ 324 + 144
→ 468
Thus,
We get required value : 468 .
Answered by
40
If α and β are the zeroes of the polynomial p(x)=x² - 5x + 6 , find the value of α⁴β² + α²β⁴.
We have,
p (x) = x² - 5x + 6
given that α and β are zeroes of p(x)
so,
also,
we have to find;
→α⁴β² + α²β⁴
taking α²β² common
→α²β² ( α² + β² )
using identity x²+y² = (x+y)²-2xy
→(αβ)² ( (α+β)²- 2αβ)
putting values of α+β & αβ using eqn(1) & (2)
→(6)² ( ( 5)² - 2 ( 6 ) )
→ 36 ( 25 - 12 )
→ 36 (13)
→ 468 (Ans.)
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