Question
Factorise:
$4 x^4+7 x^2-2$
$4 x^4+7 x^2-2$
✓
Answer
Let $x^2=y$
Then, $4 x^4+7 x^2-2$
$=4 y^2+7 y-2$
$=4 y^2+8 y-y-2$
$=4 y(y+2)-1(y+2)$
$=(y+2)(4 y-1)$
Now replacing y by $x^2$, we get
$4 x^4+7 x^2-2$
$=\left(x^2+2\right)\left(4 x^2-1\right) \text { Since } a^2-b^2=(a-b)(a+b)$
$=\left(x^2+2\right)(2 x+1)(2 x-1)$
Then, $4 x^4+7 x^2-2$
$=4 y^2+7 y-2$
$=4 y^2+8 y-y-2$
$=4 y(y+2)-1(y+2)$
$=(y+2)(4 y-1)$
Now replacing y by $x^2$, we get
$4 x^4+7 x^2-2$
$=\left(x^2+2\right)\left(4 x^2-1\right) \text { Since } a^2-b^2=(a-b)(a+b)$
$=\left(x^2+2\right)(2 x+1)(2 x-1)$
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