Q. 195.0( 1 Vote )

Factorize: x4 + x2 + 25.

Answer :

We will use completing the square method for factorizing the expression.


We have, x4 + x2 + 25


Now we can write above expression as, (x2)2 + x2 + (5)2


We need to make the middle term of the expression such that,


x2 = 2 × x2 × 5


x2 = 10 x2


So, adding and subtracting 9 x2 from the expression,


x4 + x2 + 25 = (x2)2 + x2 + (5)2 + 9 x2 – 9 x2


x4 + x2 + 25 = (x2)2 + 10 x2 + (5)2 – 9 x2


x4 + x2 + 25 = (x2 + 5)2 – 9 x2


[ Applying formula: (a + b)2 = a2 + b2 + 2 a b]


x4 + x2 + 25 = (x2 + 5)2 – (3 x)2


Now we will apply the formula,


a2 – b2 = (a + b)(a – b)


x4 + x2 + 25 = (x2 + 5 + 3 x)(x2 +5 – 3x)


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