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Factoring Sums of Cubes

Quick Overview

  • The Formula: a3+b3 = (a+b)(a2ab+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.
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Explanation of the Formula -- Direct Method

We can verify the factoring formula by expanding the result and seeing that it simplifies to the original, as follows.

(a+b)(a2ab+b2)=a2(a+b)ab(a+b)+b2(a+b)=a3+a2ba2bab2+ab2+b3=a3+a2ba2b+ab2ab2+b3=a3+0+0+b3=a3+b3

Explanation of the Formula -- Division Method

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 1

Find a solution to a3+b3=0.

a3+b3=0a3=b33a3=3b3a=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 2

Find the other factor of a3+b3 using polynomial division.

a2ab+b2a+ba3+0a2b+0ab2+b3(a3+a2b)_a2b+0ab2+b3(a2bab2)_ab2+b3(ab2+b3)_0

Since a+b divides evenly into a3+b3, we know

(a+b)(a2ab+b2)=a3+b3

Sum Of Cubes Calculator

Example

Show that x3+1 factors into (x+1)(x2x+1).

Step 1

Show that expanding (x+1)(x2x+1) results in x3+1.

(x+1)(x2x+1)=x2(x+1)x(x+1)+1(x+1)=x3+x2x2x+x+1=x3+1

Step 1 (Alternate Solution)

Show that (x+1)(x2x+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=3x3=x.

Likewise, since b is the cube root of the second term, we know b=31=1.

Step 2

Write down the factored form.

a3+b3=(a+b)(a2ab+b2)x3+1=(x+1)(x2x1+12)=(x+1)(x2x+1)

Answer

x3+1=(x+1)(x2x+1)

Continue to Practice Problems
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