(3x+y)³+(3x-y)³
X³+y³
8a⁶+729
Write a general statement on how to factorize the expressions above
To factorize the expression (3x+y)³+(3x-y)³, you can observe that it follows the sum of cubes formula, which is a³+b³ = (a+b)(a²-ab+b²). Therefore, the expression can be factorized as follows:
(3x+y)³+(3x-y)³ = [(3x+y)+(3x-y)][(3x+y)²-(3x+y)(3x-y)+(3x-y)²]
= (6x)(9x²+6xy+y²-9x²+y²)
= 6x(10x²+6xy+2y²)
For the expression x³+y³, you can see that it follows the sum of cubes formula as well. Therefore, it can be factorized as:
x³+y³ = (x+y)(x²-xy+y²)
Lastly, the expression 8a⁶+729 can be factorized as a sum of cubes, where 729 = 9³. Therefore, it can be factorized as:
8a⁶+729 = (2a²+9)(4a⁴-18a²+81)
In general, to factorize expressions involving sums of cubes or cubes plus a constant, you can look for patterns that match the sum of cubes formula (a³+b³ = (a+b)(a²-ab+b²)) and apply that formula accordingly.