if (x+a)(x+b)(x+c)=x³+14x²+59x+70 find the value. . 1.(a+b+c) 2.1/a +1/b+1/c. 3.a²+b²+c². 4.a/bc+b/ac+c/ab. please help me friends i have a exam in this I will make you as brainlist
Answers
Answer:
Answer:
(x+a)(x+b)(x +c)=x^3+14x^2+59x+70(x+a)(x+b)(x+c)=x
3
+14x
2
+59x+70
x^3+(a+b+c)x^2+(ab+bc+ca)x+abc=x^3+14x^2+59x+70x
3
+(a+b+c)x
2
+(ab+bc+ca)x+abc=x
3
+14x
2
+59x+70
Equating corresponding coefficients on both sides we get
\boxed{a+b+c=14}
a+b+c=14
.........(1)
ab+bc+ca=59ab+bc+ca=59 .......(2)
abc=70abc=70 .........(3)
\frac{(2)}{(3)}\implies\:\frac{ab+bc+ca}{abc}=\frac{59}{70}
(3)
(2)
⟹
abc
ab+bc+ca
=
70
59
\implies\:\frac{ab}{abc}+\frac{bc}{abc}+\frac{ca}{abc}=\frac{59}{70}⟹
abc
ab
+
abc
bc
+
abc
ca
=
70
59
\implies\:\frac{1}{c}+\frac{1}{a}+\frac{1}{b}=\frac{59}{70}⟹
c
1
+
a
1
+
b
1
=
70
59
\implies\:\boxed{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{59}{70}}⟹
a
1
+
b
1
+
c
1
=
70
59
we know that
(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)(a+b+c)
2
=a
2
+b
2
+c
2
+2(ab+bc+ca)
\implies\:14^2=a^2+b^2+c^2+2(59)⟹14
2
=a
2
+b
2
+c
2
+2(59)
\implies\:196=a^2+b^2+c^2+118⟹196=a
2
+b
2
+c
2
+118
\implies\:196-118=a^2+b^2+c^2⟹196−118=a
2
+b
2
+c
2
\implies\:\boxed{a^2+b^2+c^2=78}⟹
a
2
+b
2
+c
2
=78
.....(4)
\frac{(4)}{(3)}\implies\:\frac{a^2+b^2+c^2}{abc}=\frac{78}{70}
(3)
(4)
⟹
abc
a
2
+b
2
+c
2
=
70
78
\implies\:\frac{a^2}{abc}+\frac{b^2}{abc}+\frac{c^2}{abc}=\frac{39}{35}⟹
abc
a
2
+
abc
b
2
+
abc
c
2
=
35
39
\implies\:\boxed{\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab}=\frac{39}{35}}⟹
bc
a
+
ac
b
+
ab
c
=
35
39