MCQ
The factors of $x^4 + x^2 + 25$ are:
- ✓$(x^2 + 3x + 5)(x^2 - 3x + 5)$
- B$(x^2{ }+ 3x + 5)(x^2 + 3x − 5)$
- C$(x^2 + x +5)(x^2 - x + 5)$
- DNone of these.
✓
Answer
Correct option: A.
$(x^2 + 3x + 5)(x^2 - 3x + 5)$
For making perfect square to $x^4 + x^2 + 25$
We add $+10x^2$ and -$10x^2$ to it.
$= x^4 + x^2 + 25$
$= x^4 + x^2 + 25 + 10x^2 - 10x^2$
$= [x^4 + 10x^2 + 25] - 9x^2$
$= (x^2 + 5)^2 + (3x)^2$
$= [(x^2 + 5) + 3x][(x^2 + 5) - 3x]$
$= (x^2 + 3x + 5)(x^2 - 3x + 5)$
We add $+10x^2$ and -$10x^2$ to it.
$= x^4 + x^2 + 25$
$= x^4 + x^2 + 25 + 10x^2 - 10x^2$
$= [x^4 + 10x^2 + 25] - 9x^2$
$= (x^2 + 5)^2 + (3x)^2$
$= [(x^2 + 5) + 3x][(x^2 + 5) - 3x]$
$= (x^2 + 3x + 5)(x^2 - 3x + 5)$
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