2. Factoring Sums or Differences of Cubes

3. Write an equivalent expression by factoring: x3  27
Example
Write an equivalent expression by factoring: x3  27
Solution We observe that
x3  27 = x3  (3)3
In one set of parentheses, we write the first cube root, x minus the second cube root, 3:
(x  3)( )
To get the other factor, we think of x – 3 and do the following:
(x  3)(x2 + 3x + 9)
Thus, x3  27 = (x  3)(x2 + 3x + 9)
Square the first term: x2.
Multiply the terms and then change the sign: 3x.
Square the second term: (3)2, or 9.
Same
Opposite
Always Positive
= SOAP

4. Write an equivalent expression by factoring: x3 + 27
Example
Write an equivalent expression by factoring: x3 + 27
Solution We observe that
x = x3 + (3)3
In one set of parentheses, we write the first cube root, x plus the second cube root, 3:
(x + 3)( )
To get the other factor, we think of x + 3 and do the following:
(x + 3)(x2  3x + 9)
Thus, x = (x + 3)(x2  3x + 9)
Square the first term: x2.
Multiply the terms and then change the sign: 3x.
Square the second term: (3)2, or 9.
Same
Opposite
Always Positive
= SOAP

5. Example Factor: x3 + 27
Solution We have
x = x3 + (3)3 = (x + 3) (x2  x  )
A3 + B3 = (A + B) (A2  A B + B2)
The factorization is: (x + 3)(x2  3x + 9)
Example Factor: 27x3  8y3
Solution We have
27x3  8y3
= (3x)3  (2y)3
= (3x  2y)[(3x)2 + (3x)(2y) + (2y)2]
The factorization is: (3x  2y)(9x2 + 6xy + 4y2)

7. Example Factor: 3ab a4b10
Solution First we look for a greatest common factor:
3ab a4b10
= 3ab4(1 + 64a3b6)
= 3ab4[13 + (4ab2)3]
= 3ab4(1 + 4ab2)(1  4ab2 + 16a2b4)
The factorization is: 3ab4(1 + 4ab2)(1  4ab2 + 16a2b4)

8. Solving Polynomial Equations
Example: Solve x3 = 64.
Solution – Algebraic
Rewrite to get 0 on one side and factor:
x3 = 64
x3 – 64 = Subtracting 64 from both sides, set equal to zero
(x – 4)(x2 + 4x + 16) = Factoring
x2 + 4x + 16 does not factor using real coefficients.
x – 4 = or x2 + 4x + 16 = 0
x = 4
We have one solution, x = 4.
Graphical Solution
The solution is x = 4.

9. Solution – Graphical Solution
Example:
Solution – Graphical Solution
The solutions are: x = 0.070, x = 2.213, or x =