Question

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2.
If a+b+c =12 and ab + bc +ac = 22, find the
value of a3 +b3 + c3 - 3abc.​

Answer 1

Question :

If a+b+c =12 and ab + bc +ac = 22, find the value of a³ + b³ + c³ - 3abc

To find :

value of a³ + b³ + c³ - 3abc

Given Data :

→ a+b+c =12

→ ab + bc +ac = 22

Formula :

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

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

Solution :

First we need to get a² + b² + c²

So we will apply the first formula : (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

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

By putting the values we get :

• (12)² = a² + b² + c² + 2(22)

• a² + b² + c² = 144 - 44

• a² + b² + c² = 100

Now we will apply the second formula to get a³ + b³ + c³ - 3abc

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

By putting the values we get :

• (12)(100 - ab - bc - ca) = a³ + b³ + c³ - 3abc

We have to take - 1 as common to get ab + bc + ca by factorisation.

• (12)[100 - 1(ab + bc + ca)] = a³ + b³ + c³ - 3abc

• a³ + b³ + c³ - 3abc = (12)[100 - 1(22)]

• a³ + b³ + c³ - 3abc = 12(100 - 22)

• a³ + b³ + c³ - 3abc = 12 × 78

• a³ + b³ + c³ - 3abc = 936

So the value of a³ + b³ + c³ - 3abc = 936

ADDITIONAL INFORMATION :

→ Factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.

→ For example, 3 × 5 is a factorization of the integer 15, and is a factorization of the polynomial x² – 4.