without finding the zeros alpha and beta of the polynomial p(x) =3x²-10x+7 , find the value alpha + beta and alpha × beta.
Answers
Given: f(x)=3x
2
−7x−6
Which is of the from ax
2
+bx+c
⇒a=3,b=−7,c=−6
Now, α+β=−
a
b
=
3
−(−7)
α+β=
3
7
....(1)
and αβ=
a
c
=
3
−6
αβ=−2...(2)
(1) α
2
+β
2
=(α+β)
2
−2α+β
=(
3
7
)
2
−2(−2) [ from (1) & (2)]$$
=
9
49
+4=
9
49+36
=
9
85
α
2
β
2
=(αβ)
2
=(−2)
2
=4
The pohynomical whose roots are α
2
,β
2
is given by , x
2
−(sumofroots)x+productofroots
=x
2
−
9
85x
+4
=
9
9x
2
−85x+36
=
9
1
(9x
2
−85x+36)
(2) (2α+3β)+(3α+2β)=5α+5β
=5(α+β)
=5⋅
3
7
=
3
35
(2α+3β)+(3α+2β)=6α
2
4αβ+9αβ+6β
2
=6(α
2
+β
2
)+13αβ
=6{(α+β)
2
−2αβ}+Bαβ
=6(α+β)
2
−12αβ+Bαβ
=6(α+β)
2
+αβ
=6(
3
7
)
2
+(−2)
=6⋅
9
49
−2
=
3
98
−2
=
3
98−6
=
3
92
the required polynomial whose you are (2α+3β) and (3α+2β) is
x
2
−
3
35
x+
3
92
=
3
1
(3x
2
−35x+92)
Step-by-step explanation:
answer is in the attachment
