Question

If a+b+c=1,ab+bc+ac=-1 and abc=-1.Find a³+b³+c³ ...​

Answer 1

Answer:

If a+b+c=3, a²+b²+c²=5, and a³+b³ + c³=7, then what is the value of a^4+b^4 + c^4?

Here it is given that

a + b + c = 3

a² + b² + c² = 5

a³ + b³ + c³ = 7

Now because

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

3² = 5 + 2(ab + bc + ca)

2 = (ab + bc + ca)-------(1)

again,

a³ + b³ + c³ - 3abc = (a + b + c){a² + b² + c² -(ab + bc + ca)}

7 - 3abc = 3 × { 5 - (2)} from equation (1)

7 - 3abc = 9

-3abc = 9 — 7 = 2

abc = - 2/3---- - - - - - (2)

now,

a⁴ + b⁴ + c⁴ = {a² + b² + c² }² - 2{a²b² + b²c² + c²a² }

= (5)² - 2{(ab + bc + ca) ² - 2(ab.bc +bc.ca+ ca.ab)}

= 25 - 2{ (2)² - 2abc(a + b +c)}

= 25 - 2{ 4 —2 × (-2/3) × (3)}

= 25 - 2{ 4 + 4 }

= 25 - 16

= 9