Answer should be in the form of a3 + b3

Question

asked Feb 14, 2023 157k views

1 Answer

Kavinyao · Answer 1 · 2023-02-18T21:28:35+0000

Answer:

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
from first parentheses:

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