Verify: (a + b + c) (a² + b² + c² - ab - bc - ca) = a³ + b³+ c³ - 3abc
Solution:
Given, (a + b + c) (a² + b² + c² - ab - bc - ca) = a³ + b³+ c³ - 3abc
Here, L.H.S = (a + b + c) (a² + b² + c² - ab - bc - ca)
= [a³ + ab² + ac² - a²b - abc - ca²] + [a²b + b³ + bc² - ab² - b²c - abc] + [ca² + cb² + c³ - abc - bc² - c²a]
Sorting,
= a³ + b³ + c³ + (ab² - ab²) + (ac² - c²a) + (a²b - a²b) + (ca² - ca²) + (bc² - bc²) +(cb² - b²c) - abc - abc - abc
= a³ + b³+ c³ - 3abc
= R.H.S
✦ Try This: Verify (a + b + c) (a²+ b² + c² - ab - bc - ca) = a³ + b³ + c³ - 3abc if a = 3, b = 2 and c = 5
Given, (a + b + c) (a² + b² + c² - ab - bc - ca) = a³ + b³+ c³ - 3abc
L.H.S = (a + b + c) (a² + b² + c² - ab - bc - ca)
Substituting a = 3, b = 2 and c = 5
L.H.S = (3 + 2 + 5) [(3)² + (2)² + (5)² - (3 × 2) - (2 × 5) - (5 × 3)]
= 10 [9 + 4 + 25 - 6 - 10 - 15]
= 10 [7] = 70
R.H.S = a³ + b³+ c³ - 3abc
= (3)³ + (2)³ + (5)³ - (3 × 3 × 2 × 5)
= 27 + 8 + 125 - 90
= 70
L.H.S = R.H.S
☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 9
NCERT Exemplar Class 8 Maths Chapter 7 Problem 113(ii)
Verify: (a + b + c) (a² + b² + c² - ab - bc - ca) = a³ + b³+ c³ - 3abc
Summary:
(a + b + c) (a² + b² + c² - ab - bc - ca) = a³ + b³+ c³ - 3abc is true
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