if alpha, beta ate the roots of the equation x^2+2x-15=0, find alpha square +beta square, alpha cube + beta cube
Answers
Given polynomial = x² - 2x - 15
The given polynomial's zero are = α and β
The new polynomial when zeros are 2α and 2β = ??
In polynomial :
a = 1
b = -2
c = -15
____________________________
Sum of zeros:
____________________________
Product of zeros:
____________________________
For the new polynomial,
The sum of zeros for new polynomial is 4.
The product of zeros for new polynomial is -60.
____________________________
The new polynomial will be =
Therefore, the new polynomial is x² - 4x - 60
Answer- x
2
+2x+2=0
Here, a=1,b=2,,c=2
From quadratic formula,
x=
2×1
−2±
2
2
−4×2×1
⇒x=
2
−2±2i
=−1±i
Therefore,
α=−1+i⇒α
2
=(−1+i)
2
=−2i
β=−1−i⇒β
2
=(−1−i)
2
=2i
Now,
α
15
+β
15
=(α
2
)
7
⋅α+(β
2
)
7
⋅β
=(−2i)
7
(−1+i)+(2i)
7
(−1−i)
=(−2)
7
(−i)(−1+i)+2
7
(−i)(−1−i)
=2
7
i(−1+i)+2
7
i(1+i)
=2
7
i(−1+i+1+i)
=2
7
i⋅2i
=2
8
⋅(−1)
=−256