Question
Evaluate: $(a+1)\left(a^2-a+1\right)$ and $(a-1)\left(a^2+a+1\right)$
✓
Answer
$(a+1)\left(a^2-a+1\right)$ and $(a-1)\left(a^2+a+1\right)$
$= a (a^2 - a + 1) + 1 (a^2 - a + 1)$
$= a^3 - a^2 + a + a^2 - a + 1$
$= a^3 + 1$
$(a - 1)(a^2 + a + 1)$
$= a(a^2 + a + 1) - 1(a^2 + a + 1)$
$= a^3 + a^2 + a - a^2 - a - 1$
$= a^3 - 1$
Now, $(a + 1)(a^2 - a + 1) + (a - 1)(a^2 + a + 1)$
$= a^3 + 1 + a^3 - 1$
$= 2a^3$
$= a (a^2 - a + 1) + 1 (a^2 - a + 1)$
$= a^3 - a^2 + a + a^2 - a + 1$
$= a^3 + 1$
$(a - 1)(a^2 + a + 1)$
$= a(a^2 + a + 1) - 1(a^2 + a + 1)$
$= a^3 + a^2 + a - a^2 - a - 1$
$= a^3 - 1$
Now, $(a + 1)(a^2 - a + 1) + (a - 1)(a^2 + a + 1)$
$= a^3 + 1 + a^3 - 1$
$= 2a^3$
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