Expand the following: 3 x + 1 3 x 3

To find: 3 x + 1 3 x 3 Using a + b 3 = a 3 + b 3 + 3 a 2 b + 3 a b 2 Here, a = 3 x & b = 1 3 x ⇒ 3 x + 1 3 x 3 = 3 x 3 + 1 3 x 3 + 3 3 x 2 1 3 x + 3 3 x 1 3 x 2 = 27 x 3 + 1 27 x 3 + 9 x + 1 x Therefore, 3 x + 1 3 x 3 = 27 x 3 + 1 27 x 3 + 9 x + 1 x

If x 2 - 9 x + 1 = 0 , find the value of x + 1 x and x 3 + 1 x 3 .

We need to find the value of x + 1 x and x 3 + 1 x 3 . Given that, x 2 - 9 x + 1 = 0 We can write the expression as, x 2 + 1 = 9 x (On dividing both sides by x ), we get x + 1 x = 9 Taking cube on both sides, x + 1 x 3 = 9 3 Using identity, a + b 3 = a 3 + b 3 + 3 a 2 b + 3 a b 2 x 3 + 1 x 3 + 3 × x × 1 x × x + 1 x = 729 ⇒ x 3 + 1 x 3 + 3 9 = 729 ⇒ x 3 + 1 x 3 = 702 Hence the value of x + 1 x = 9 and x 3 + 1 x 3 = 702