Question

Answer should be in the form of a3 + b3 ​

Answer 1

Question:

`\textsf{factor : \ } x^4+(1)/(x^2)`

Rewrite the expression using the exponent rule
`(a^b)/(a^c)=a^((b-c))`

`\implies x^4+(1)/(x^2)=(x^6)/(x^2)+(1)/(x^2)`

Factor out common term
`(1)/(x^2)` :

`\implies (1)/(x^2)(x^6+1)}`

Factor
`(x^6+1)`:

Apply the exponent rule
`(a^b)^c=a^(bc)` to get cubic exponents:

`\implies x^6+1 = (x^2)^3+1^3`

Apply the sum of cubes formula:
`a^3+b^3=(a+b)(a^2-ab+b^2)`

where
`a=x^2` and
`b=1`:

`\implies (x^2)^3+1^3=(x^2+1)((x^2)^2-x^2 \cdot 1+1^2)`

`\implies (x^2+1)(x^4-x^2+1)`

Therefore,

`\implies (1)/(x^2)(x^6+1)}=(1)/(x^2)(x^2+1)(x^4-x^2+1)`

Simplify by dividing the 2nd parentheses by
`x^2`:

`\implies (x^2+1)\left((x^4-x^2+1)/(x^2)\right)`

`\implies (x^2+1)\left(x^2-1+(1)/(x^2)\right)`

Factor out
`x` from first parentheses:

`\implies x\left(x+(1)/(x)\right)\left(x^2-1+(1)/(x^2)\right)`