1. Factorization by Using Identities of Sum and Difference of Two Cubes

2. If cube B is taken out from cube A to form solid C, what is the volume of solid C?
What are the volumes of the cube A and cube B?
Volume of solid C = a3 – b3
Volume of cube A = a3
Volume of cube B = b3

3. Can you find the volumes of cuboids X, Y and Z respectively?
Suppose solid C divided into 3 cuboids X, Y and Z.
Volume of cuboid X = a2(a – b)
Volume of cuboid Y = ab(a – b)
Volume of cuboid Z = b2(a – b)

4. ∵ ∵ volume of solid C = volume of cuboid X +
volume of cuboid Y + volume of cuboid Z ∵
a3 – b3 = a2(a – b) + ab(a – b) + b2(a – b)
= (a – b)(a2 + ab + b2)
By substituting b = -c into the expression, we have:
a3 + c3 = (a + c)(a2 – ac + c2)

5. As a result, we have two algebraic identities:
They are called the identities of the sum and the difference of two cubes.

6. Factorize the following expressions.
(a) y3 + 8
(b) 64z3 – 1

7. Follow-up question Factorize the following expressions.
(a) 1 – x3 (b) s3 + 8t3
Solution