6. The number of polynomials having zeroes as 5 and -3 is (a) (b)2 (C)3 (d) more than 3
Answers
Let p(x) be the required polynomial, in which zeroes of the polynomial are -2 and 5.
The polynomial is of the form,
p
(x)=a
x 2
+bx+c
p(x)=ax2+bx+c
To find the variables a and b:
Sum of the zeroes =
−b
a
−ba
, where
b is coefficient of
x
x
a is coefficient of
x 2
x2
−2+5=
−b
a
−2+5=−ba
3=
−b
a
3=−ba
Hence,
3
1
=
−b
a
31=−ba
Therefore, the values of a and b is
a=1
a=1
and
b=−3
b=−3
.
Product of the zeroes = constant term ÷ coefficient of x2 i.e.
Product of zeroes =
c
a
ca
, where
c is the constant term.
a is coefficient of
x 2
x2
.
(
−2)5=
c
a
(−2)5=ca
−10=
c
a
−10=ca
Hence,
−10
1
=
c
a
−101=ca
Therefore, the value of c is
−10=c
−10=c
Hence, we got
a=1
a=1
,
b=−3
b=−3
and
c=−10
c=−10
.
Substituting the values of a, b and c in the polynomial
p
(x)=a
x 2
+bx+c
p(x)=ax2+bx+c
p
(x)=1⋅
x 2
−3x−10
p(x)=1⋅x2−3x−10
Therefore, p(x) is
p
(x)=
x 2
−3x−10
p(x)=x2−3x−10
Therefore, we can conclude that x can take any value.
Hence, option D is the correct answer.
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