How to factor the difference of cubes
The formula we use for factoring the difference of cubes
In this lesson we’ll look at how to recognize a difference of two cubes and then use a formula to factor it.
We know we’re dealing with the difference of cubes, because we have two perfect cubes separated by subtraction.
When that’s the case, we can take the cube (third) root of each term.
The formula for a difference of cubes is
???a^3-b^3=(a-b)(a^2+ab+b^2)???
When we have the difference of cubes and how to factor it
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Factoring the difference of cubes by identifying a and b
Example
Factor the polynomial.
???c^3-8b^{12}???
If we check to see whether either term is a cube,
???\sqrt[3]{c^3}=c???
???\sqrt[3]{8b^{12}}=2b^4???
we can see that both terms are perfect cubes. The difference of cubes formula says ???a^3-b^3??? is always factored as
???(a-b)(a^2+ab+b^2)???
Since in this case ???a=c??? and ???b=2b^4???, we get
???(c-2b^4)\left(c^2+c(2b^4)+(2b^4)^2\right)???
???(c-2b^4)(c^2+2b^4c+4b^8)???
Let’s do one more.
Example
Factor the expression.
???27x^3y^9-216z^{15}???
If we check to see whether either term is a cube,
???\sqrt[3]{27x^3y^9}=3xy^3???
???\sqrt[3]{216z^{15}}=6z^5???
we can see that both terms are perfect cubes. The difference of cubes formula says ???a^3-b^3??? is always factored as
???a^3-b^3=(a-b)(a^2+ab+b^2)???
The variable ???a??? will be the cube root of the first term, and the variable ???b??? will be the cube root of the second term. So
???a=3xy^3???
???b=6z^5???
The formula gives us
???a^3-b^3=(a-b)(a^2+ab+b^2)???
???(3xy^3-6z^5)\left((3xy^3)^2+(3xy^3)(6z^5)+(6z^5)^2\right)???
???(3xy^3-6z^5)(9x^2y^6+18xy^3z^5+36z^{10})???
We can check our work by distributing each term in the binomial factor over each term in the trinomial factor.
???(3xy^3)(9x^2y^6)+(3xy^3)(18xy^3z^5)+(3xy^3)(36z^{10})???
???+(-6z^5)(9x^2y^6)+(-6z^5)(18xy^3z^5)+(-6z^5)(36z^{10})???
???27x^3y^9+54x^2y^6z^5+108xy^3z^{10}-54x^2y^6z^5-108xy^3z^{10}-216z^{15}???
???27x^3y^9+54x^2y^6z^5-54x^2y^6z^5+108xy^3z^{10}-108xy^3z^{10}-216z^{15}???
???27x^3y^9-216z^{15}???