How to factorize x^4 + 5x^2 + 9 ?

x⁴ + 5x² + 9

= x⁴ + 5x² + 9 + x² - x², adding and subtracting x²

= x⁴ + 5x² + x² + 9 - x², using commutativity

= (x⁴ + 6x² +9) - x² , adding 5x² and x²

= [ (x²)² + 2.x².3 + 3² ] - x²,  using (A^m)ⁿ = A^mn and factorizing 6

= (x² + 3) ² - x², using (A + B)² = A² + 2AB + B²

= (x² + 3 - x)(x² + 3 + x), using (A - B)² = (A - B)(A + B)

= (x² - x + 3)(x² + x +3), Ans. , using commutativity

How to factorize 9x^2 - 4a^2 + 4ay - y^2

9x² - 4a² + 4ay - y²
= 9x² - (4a² - 4ay + y²), taking ' - ' common
= (3x)² - (2a - y) ² , since A² - 2AB + B² = (A - B)²
= [3x - (2a -y)][3x + (2a - y)], using A² - B² = (A - B)(A + B)
= (3x - 2a + y)(3x + 2a - y) Ans. , using commutativity

Factorization of (x - 2)(x + 2) + 3

(x - 2)(x + 2) + 3
= (x^2 - 2^2) + 3, since (A -B)(A + B) = A^2 - B^2
=  x^2 - 4 + 3
= x^2 - 1
= x^2 - 1^2, since 1^2 = 1
= (x - 1)(x + 1) Ans. using A^2 - B^2 = (A -B)(A + B)

How to factorize x^2 + 1/x^2 - 11

x^2 + 1/x^2 - 11

= x^2 + 1/x^2 - 9 - 2

= (x^2 - 2 + 1/x^2) - 9 , using commutativity and associativity

= (x^2 - 2x * 1/x + 1/x^2) - 3^2, since x * 1/x = 1

= (x - 1/x)^2 - 3^2, using A^2 -2AB + B^2 = (A - B)^2

= (x - 1/x + 3)(x - 1/x -3), Ans. using A^2 - B^2 = (A + B)(A - B)

Factorization of x^4 + 4

x^4 + 4
= x^4 + 4x^2 + 4 - 4x^2, adding and subtracting 4x^2 (the value remains unchanged)
= (x^4 + 4x^2 + 4) - 4x^2
= (x^2 + 2)^2 - 4x^2, since (A + B)^2 = A^2 + B^2
= (x^2 + 2)^2 - (2x)^2
= (x^2 + 2 -2x)(x^2 + 2 +2x), using (A^2 - B^2) = (A - B)(A + B)
= (x^2 - 2x + 2)(x^2 + 2x +2) Ans. , using commutativity

Factorize 3x^5 - 48x

3x^5 - 48x
= 3x(x^4 - 16), taking 3x common
= 3x[(x^2)^2 - 4^2]
= 3x[(x^2 - 4)(x^2 + 4)], since A^2 - B^2 = (A - B)(A + B)
= 3x[(x^2 - 2^2)(x^2 + 4)]
= 3x[(x - 2)(x + 2)(x^2 + 4)], using, again, A^2 - B^2 = (A - B)(A + B)
= 3x(x - 2)(x + 2)(x^2 + 4) Ans.

How to factorize x^4 + 3x^2 + 4

Since 3x^2 can be written as 4x^2 - x^2, we can write

x^4 + 3x^2 + 4 = x^4 + 4x^2 + 4 - x^2

= (x^2 + 2) ^ 2 - (x)^2 , using (A + B)^2 = A^2 + 2AB + B^2 where A = x^2 and B = 2

= (x^2 + 2 + x) (x^2 + 2 - x) , using A^2 - B^2 = (A + B)(A - B)

= (x^2 + x + 2) (x^2 - x + 2)  Ans., using commutativity