If a + b + c = 8 and a b + b c + c a = 20 , find the value of a 3 + b 3 + c 3 – 3 a b c . Options: A . &nbsp16 B . &nbsp32 C . &nbsp48 D . &nbsp64

If a+b+c=8 and ab+bc+ca=20, find the value of a3+b3+c3

Question

a + b + c = 8

and

a b + b c + c a = 20

, find the value of

a^{3} + b^{3} + c^{3} - 3 a b c

Options:

A . 16

B . 32

C . 48

D . 64

Answer: Option B
:
B
Since

{(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)

a + b + c = 8 and a b + b c + c a = 20

{(8)}^{2} = a^{2} + b^{2} + c^{2} + 2 \times 20

64 = a^{2} + b^{2} + c^{2} + 40

a^{2} + b^{2} + c^{2} = 24

We now use the following identity:

a^{3} + b^{3} + c^{3} - 3 a b c = (a + b + c) (a^{2} + b^{2} + c^{2} - (a b + b c + c a))

a^{3} + b^{3} + c^{3} - 3 a b c = 8 \times (24 - 20) = 4 \times 8 = 32

Thus,

a^{3} + b^{3} + c^{3} - 3 a b c = 32

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