Find the zeroes of each of
the following polynomial
and verify
Retionship between the
zeroes and their derfficients
a x² - 90+20
Answers
Step-by-step explanation:
their coefficients :
(i) f(x)=x
2
−2x−8
(ii) g(x)=4s
2
−4s+1
(iii) f(x)=x
2
−(
3
+1)x+
3
(iv) x
2
−3−7x
(v) p(x)=x
2
+2
2
x−6
(vi) q(x)=
3
x
2
+10x+7
3
(vii) g(x)=a(x
2
+1)−x(a
2
+1)
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ax
2
+bx+c=0⇒α+β=−
a
b
,αβ=
a
c
(i)
x
2
−2x−8=0
⇒a=1,b=−2,c=−8
x
2
−2x−8=0
(x−4)(x+2)=0
α=−2,β=4
α+β=−
a
b
→−2+4=−
1
−2
⇒2=2
αβ=
a
c
→(−2)(4)=
1
−8
⇒−8=−8
Hence the relationship between zeros and coefficients is verified.
(ii)
4s
2
−4s+1=0
⇒a=4,b=−4,c=1
4s
2
−4s+1=0
(2s−1)(2s−1)=0
α=
2
1
,β=
2
1
α+β=−
a
b
→
2
1
+
2
1
=−
4
−4
⇒1=1
αβ=
a
c
→
2
1
×
2
1
=
4
1
⇒
4
1
=
4
1
Hence the relationship between zeros and coefficients is verified.
(iii)
x
2
−(
3
+1)x+
3
=0
⇒a=1,b=−(
3
+1),c=
3
x
2
−(
3
+1)x+
3
=0
x=
2⋅1
−(−
3
−1)±
(−
3
−1)
2
−4⋅1
3
=
3
,1
α=
3
,β=1
α+β=−
a
b
→
3
+1=−
1
−(
3
+1)
⇒LHS=RHS
αβ=
a
c
→1×
3
=
1
3
⇒LHS=RHS
Hence the relationship between zeros and coefficients is verified.
(iv)
x
2
−7x−3=0
⇒a=1,b=−7,c=−3
x
2
−7x−3=0
x=
2⋅1
−(−7)±
(−7)
2
−4⋅1(−3)
:
2
7±
61
α=
2
7+
61
,β=
2
7−
61
α+β=−
a
b
→
2
7+
61
+
2
7−
61
=−
1
−7
⇒LHS=RHS
αβ=
a
c
→
2
7+
61
×
2
7−
61
=
1
−3
⇒LHS=RHS
Hence the relationship between zeros and coefficients is verified.
(v)
x
2
+2
2
x−6=0
⇒a=1,b=2
2
,c=−6
x
2
+2
2
x−6=0
x=
2⋅1
−2
2
±
(2
2
)
2
−4⋅1(−6)
=
2
,−3
2