Example 1 · level 1
If x + 1/x = 7, find the value of x² + 1/x².
Show solution
- (x + 1/x)² = x² + 1/x² + 2.
- 49 = x² + 1/x² + 2.
- x² + 1/x² = 47.
Answer: 47
CLASS 9 MATHS · POLYNOMIALS
Algebraic identities (squares and cubes)
x² + 1/x² = (x + 1/x)² − 2; a³ + b³ = (a + b)³ − 3ab(a + b); if a + b + c = 0 then a³ + b³ + c³ = 3abc.
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Worked examples
Example 2 · level 2
Without finding the cubes, evaluate (−6)³ + 3³ + 3³.
Show solution
- −6 + 3 + 3 = 0.
- So the sum of cubes = 3 × (−6) × 3 × 3
- = −162.
Answer: -162
Example 3 · level 3
If a + b = 13 and ab = 40, find a³ + b³.
Show solution
- a³ + b³ = (a + b)³ − 3ab(a + b)
- = 2197 − 3 × 40 × 13 = 2197 − 1560
- = 637.
Answer: 637