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.
Hi! I'm krista.
I create online courses to help you rock your math class. Read more.
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
Take the course
Want to learn more about Algebra 2? I have a step-by-step course for that. :)
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.
We know we’re dealing with the difference of cubes, because we have two perfect cubes separated by subtraction.
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}???
