Quick Overview
- The Formula: a3+b3 = (a+b)(a2−ab+b2)
- This factoring works for any binomial that can be written as a3+b3
- The discussion below includes 2 ways of demonstrating that this formula is valid.
We can verify the factoring formula by expanding the result and seeing that it simplifies to the original, as follows.
(a+b)(a2−ab+b2)=a2(a+b)−ab(a+b)+b2(a+b)=a3+a2b−a2b−ab2+ab2+b3=a3+a2b−a2b+ab2−ab2+b3=a3+0+0+b3=a3+b3
Another way to confirm to the formula is to find a solution to a3+b3=0, and then use division to find the factored form.
Step 1Find a solution to a3+b3=0.
a3+b3=0a3=−b33√a3=3√−b3a=−b
One of the solutions is a=−b. Adding b to both sides of this equation gives us a+b=0, which means (a+b) is a factor of a3+b3.
Step 2Find the other factor of a3+b3 using polynomial division.
a2−ab+b2a+ba3+0a2b+0ab2+b3−(a3+a2b)_−a2b+0ab2+b3−(−a2b−ab2)_ab2+b3−(ab2+b3)_0
Since a+b divides evenly into a3+b3, we know
(a+b)(a2−ab+b2)=a3+b3
Show that x3+1 factors into (x+1)(x2−x+1).
Step 1Show that expanding (x+1)(x2−x+1) results in x3+1.
(x+1)(x2−x+1)=x2(x+1)−x(x+1)+1(x+1)=x3+x2−x2−x+x+1=x3+1
Step 1 (Alternate Solution)Show that (x+1)(x2−x+1) matches the correct pattern for the formula.
Since we want to factor x3+1, we first identify a and b.
Since a is the cube root of the first term, we know a=3√x3=x.
Likewise, since b is the cube root of the second term, we know b=3√1=1.
Step 2Write down the factored form.
a3+b3=(a+b)(a2−ab+b2)x3+1=(x+1)(x2−x⋅1+12)=(x+1)(x2−x+1)
Answerx3+1=(x+1)(x2−x+1)