If a and B are zeroes of p(x) = x2 - 2x + 3, find a polynomial whose zeroes are
a-1/a+1 and B-1/B+1
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Answer:
Given that
a and b are the zeros of
x
2
−2x+3
Then,
We know that
Sum of zeros =−
coeff. of x
2
coeff. of x
a+b=−
+1
−2
a+b=2−−−−−−−−(1)
Now,
Product of zeros =
coeff of x
2
constant term
a.b=
1
3
ab=3−−−−−−−−−−−(2)
If
a+1
a−1
and
b+1
b−1
are zeros of other polynomial
Then,
Sum of zeros =
a+1
a−1
+
b+1
b−1
=
(a+1)(b+1)
(a−1)(b+1)+(b−1)(a+1)
=
ab+a+b+1
ab−b+a−1+ab+b−a−1
=
ab+(a+b)+1
2ab−2
=
3+(2)+1
2×3−2
=
6
4
Sum of zeros =
3
2
Now,
Product of zeros =(
a+1
a−1
)(
b+1
b−1
)
=
ab+a+b+1
ab−a−b+1
=
3+(a+b)+1
3−(a+b)+1
=
3+2+1
3−2+1
=
6
2
=
3
1
Product of zeros
3
1
So, equation of polynomial
x
2
− (sum of zeros) x+ product of zeros =0
x
2
−
3
2
x+
3
1
=0
3x
2
−2x+1=0
Hence, this is the answer.
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