#### Solve simultaneous linear equations:

(i) x^{3} + x^{2} – (1/x^{2}) + (1/x^{3})

(ii) (x + 1)^{6} – (x – 1)^{6}

**Solution:**

(i) x^{3} + x^{2} – (1/x^{2}) + (1/x^{3})

Rearranging the above terms, we get,

x^{3} + (1/x^{3}) + x^{2} – (1/x^{2})

We know that, a^{3} – b^{3} = (a – b) (a^{2} + ab + b^{2}) and (a^{2} – b^{2}) = (a + b) (a – b)

(x + 1/x) (x^{2} – 1 + 1/x^{2}) + (x + 1/x) (x – 1/x)

(x + 1/x) [x^{2} – 1 + 1/x^{2} + x – 1/x]

(ii) (x + 1)^{6} – (x – 1)^{6}

(x + 1)^{6} – (x – 1)^{6}

Above terms can be written as,

((x + 1)^{3})^{2} – ((x – 1)^{3})^{2}

We know that, (a^{2} – b^{2}) = (a + b) (a – b)

[(x + 1)^{3} + (x – 1)^{3}] [(x + 1)^{3} – (x – 1)^{3}]= [(x + 1) + (x – 1)][(x + 1)^{2} – (x – 1) (x + 1) + (x – 1)^{2}] [(x + 1) – (x – 1)][(x + 1)^{2} + (x – 1) (x + 1) + (x – 1)^{2}]

(x + 1 + x – 1) [x^{2} + 2x + 1 – x^{2} + 1 + x^{2} + 1 – 2x(x + 1) – x + 1] [x^{2} + 2x + 1 + x^{2} – 1 + x^{2} – 2x + 1]

By simplifying we get,

2x(x^{2} + 3) 2(3x^{2} + 1)

4x(x^{2} + 3) (3x^{2} + 1)

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