W 2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two time, and the product of its zeroes as 2.-7, -14 respectively. These exercises are not from the examination point of view?
Answers
Answer: Let p(x)=x
3
+3x
2
−x−3
p(1)=(1)
3
+3(1)
2
−1−3=0
p(−1)=(−1)
3
+3(−1)
2
+1−3=0
p(−3)=(−3)
3
+3(−3)
2
+3−3=0
Hence, 1,−1 and −3 are the zeroes of the given polynomial.
If α,β,γ, are roots of a cubic equation ax
3
+bx
2
+cx+d=0, then
1. α+β+γ=−
a
b
2. α×β+γ×β×γ+α×γ=
a
c
3. α×β×γ=−
a
d
⇒−3=−
a
b
−3=−
1
3
=−3
⇒−1=−
1
1
−1=−1
⇒3=−
1
(−3)
3=3
Hence the relationship between zeroes and coefficients is also satisfied.
Let p(x)=x
3
+3x
2
−x−3
p(1)=(1)
3
+3(1)
2
−1−3=0
p(−1)=(−1)
3
+3(−1)
2
+1−3=0
p(−3)=(−3)
3
+3(−3)
2
+3−3=0
Hence, 1,−1 and −3 are the zeroes of the given polynomial.
If α,β,γ, are roots of a cubic equation ax
3
+bx
2
+cx+d=0, then
1. α+β+γ=−
a
b
2. α×β+γ×β×γ+α×γ=
a
c
3. α×β×γ=−
a
d
⇒−3=−
a
b
−3=−
1
3
=−3
⇒−1=−
1
1
−1=−1
⇒3=−
1
(−3)
3=3
Hence the relationship between zeroes and coefficients is also satisfied.
Step-by-step explanation: