Question

8. The sum of three numbers is 3. The sum of their squares is 5 while the sum of their cubes is 9. Find the product of the numbers.​

Answer 1

Step-by-step explanation:

$\pmb{ \text{ \( \tt a+b+c=3, a^{2}+b^{2}+c^{2}=5 \) and \( \tt a^{3}+b^{3}+c^{3}=9 . \)}}$

$\pmb{ \text{Now, \( \tt a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right) . \)}}$

$\tt\therefore \quad 9-3 a b c=3(5-a b-b c-c a) .$

$\pmb{ \text{ But \( \tt \quad(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(a b+b c+c a) . \)}}$

$\pmb{ \tt\therefore \quad 3^{2}=5+2(a b+b c+c a) . }$

$\pmb{ \tt\therefore \quad a b+b c+c a=2 \qquad \: Use (1).}$