Help with factoring polynomials

Question

how do I factor these polynomials:

 

 

1.  a^4 + a^2

 

2.  8x^3 - 2x

 

3.  8a^3 - 27

Answer 1

1.  factor out the common factor, a^2(a^2+1)

 

2.  same strategy:  2x(4x^2-1)

 

3.  difference of perfect cubes   Step #1:  2^3a^3-3^3      Step #2:  (2a-3)(4a^2+6a+9)

Answer 2

1.  Look for a common factor.  a^4 = (a^2)(a^2) and a^2 = 1(a^2)

 

Therefore a^4 + a^2 = (a^2)(a^2) + 1(a^2) = (a^2)(a^2 + 1).

 

2.  Again, common factor: 2x(4x^2 - 1).   4x^2 - 1 is a difference of squares; (2x-1)(2x+1),

 

Putting these together, we get:  8x^3 - 2x = 2x(2x-1)(2x+1).

 

3.  Difference of cubes:  (x^3 - y^3) = (x - y)(x^2 + xy +y^2);

   Here x = 2a and y = 3.   The rest is left to the student.

 

I hope this helps.

 

Kevin