How do you factor #(a+b)^6 - (a-b)^6#?

Question

Eva Adamson · Accepted Answer

#(a+b)^6-(a-b)^6=4ab(3a^2+b^2)(a^2+3b^2)#

The difference of squares identity can be written:

#x^2-y^2=(x-y)(x+y)#

The difference of cubes identity can be written:

#x^3-y^3=(x-y)(x^2+xy+y^2)#

The sum of cubes identity can be written:

#x^3+y^3=(x+y)(x^2-xy+y^2)#

Hence:

#x^6-y^6#

#=(x^3)^2-(y^3)^2#

#=(x^3-y^3)(x^3+y^3)#

#=(x-y)(x^2+xy+y^2)(x+y)(x^2-xy+y^2)#

Now let #x=a+b# and #y=a-b# to find:

#(a+b)^6-(a-b)^6#

#= x^6-y^6#

#=(x-y)(x^2+xy+y^2)(x+y)(x^2-xy+y^2)#

#=((a+b)-(a-b))((a+b)^2+(a+b)(a-b)+(a-b)^2)((a+b)+(a-b))((a+b)^2-(a+b)(a-b)+(a-b)^2)#

#=(2b)(a^2+color(red)(cancel(color(black)(2ab)))+color(red)(cancel(color(black)(b^2)))+a^2-color(red)(cancel(color(black)(b^2)))+a^2-color(red)(cancel(color(black)(2ab)))+b^2)(2a)(color(red)(cancel(color(black)(a^2)))+color(red)(cancel(color(black)(2ab)))+b^2-color(red)(cancel(color(black)(a^2)))+b^2+a^2-color(red)(cancel(color(black)(2ab)))+b^2)#

#=4ab(3a^2+b^2)(a^2+3b^2)#

Eva Adamson · Answer

undefined The factored form of $ (a+b)^6 - (a-b)^6 $ is $ 64ab(a^2 + 3b^2)(a^2 - b^2) $

Isaiah Anthony · Answer

undefined To factor the expression (a+b)^6 - (a-b)^6, we can use the formula for the difference of two cubes, which states that $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$. Applying this formula to our expression, we get:

$(a+b)^6 - (a-b)^6 = [(a+b)^3 - (a-b)^3][(a+b)^3 + (a-b)^3]$.

Now, we can further factor $(a+b)^3 - (a-b)^3$ using the same formula for the difference of cubes:

$(a+b)^3 - (a-b)^3 = (a+b - (a-b))[(a+b)^2 + (a+b)(a-b) + (a-b)^2] = 2b[(a+b)^2 + ab - ab + (a-b)^2] = 2b[(a+b)^2 + (a-b)^2]$.

Similarly, we can factor $(a+b)^3 + (a-b)^3$:

$(a+b)^3 + (a-b)^3 = (a+b + (a-b))[(a+b)^2 - (a+b)(a-b) + (a-b)^2] = 2a[(a+b)^2 - ab + ab + (a-b)^2] = 2a[(a+b)^2 + (a-b)^2]$.

Now, substituting these results back into the original expression, we get:

$[(a+b)^3 - (a-b)^3][(a+b)^3 + (a-b)^3] = 2b[(a+b)^2 + (a-b)^2] \times 2a[(a+b)^2 + (a-b)^2] = 4ab[(a+b)^2 + (a-b)^2]$.

Therefore, the factored form of the expression $ (a+b)^6 - (a-b)^6 $ is $ 4ab[(a+b)^2 + (a-b)^2] $.