9. the zeroes of the polynomial F(x)=x3 - 3x2 + 1 are a-b ,a,a+b,find a and b
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CORRECT QUESTION:
The zeroes of the polynomial f(x)=x³ - 3x²+x + 1 are a-b ,a,a+b,find a and b
GIVEN:
f(x)=x³-3x²+x+1
Let α,β and γ be the zeroes of the polynomial,then,
a-b=α
a=β
a+b=γ
TO FIND:
The values of a and b
SOLUTION:
Sum of zeroes of a cubic polynomial=(-coefficient of x²/coefficient of x³)
α+β+γ=[-(-3)]/1
(a-b)+a+(a+b)=3 (∵ +b and -b gets cancelled)
3a=3⇒a=1
∴a=1
Now,
Product of zeroes of a cubic polynomial= -Constant term/coefficient of x³
αβγ= -1/1
(a-b)a(a+b)= -1
(a-b)(a+b)a= -1
(a²-b²)a= -1
(1²-b²)1= -1
1-b²= -1
√b²=√2⇒b=±√2
∴b=±√2
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