what does factoring the sum or difference of two cubes mean
It means using the handy known factorizations, as in
(a^3+b^3) = (a+b)(a^2-ab+b^2)
(a^3-b^3) = (a-b)(a^2+ab+b^2)
Of course, only the sum is needed, since
(a^3-b^3) = (a^3+(-b)^3)
once you recognize the sum of two cubes, like
125x^6+216/z^9
where
a = 5x^2
b = 6/z^3
just plug into the formula
125x^6 + 216/z^9 = (5x^2+6/z)(25x^4-30x^2/z^3+36/z^6)
so, it helps to know your cubes of small numbers, and watch for exponents which are multiples of 3.
Factoring the sum or difference of two cubes refers to the process of rewriting an algebraic expression that represents the sum or difference of two cubed terms as a product of lower-degree terms. This concept is derived from the formulas for the sum and difference of cubes.
The sum of two cubes formula states that:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
This formula states that the sum of two cubes can be factored into a binomial (a + b) multiplied by a trinomial (a^2 - ab + b^2).
The difference of two cubes formula states that:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
This formula states that the difference of two cubes can be factored into a binomial (a - b) multiplied by a trinomial (a^2 + ab + b^2).
To factor the sum or difference of two cubes, follow these steps:
Step 1: Identify the expression as a sum or difference of cubes.
Step 2: Determine the values of a and b, which are the cube roots of the terms being cubed. For example, if you have (x^3 + 8), a = x and b = 2 because x^3 = (x)^3 and 8 = (2)^3.
Step 3: Apply the appropriate formula based on whether it is a sum or difference of cubes.
Step 4: Substitute the values of a and b into the formula to obtain the factored form.
Step 5: Simplify and rearrange the factored form as needed.
By using these steps, you can factor the sum or difference of two cubes and express them as a product of lower-degree terms.