Lesson 3: Sum and Difference of Cubes — Practice Questions

1. 1. Factor the expression $x^3 + 64$. The trinomial factor is ___.

2. 2. Which of the following is the correct factorization of the expression $27m^3 - 8n^3$?

   • A. $(3m - 2n)(9m^2 + 6mn + 4n^2)$
   • B. $(3m - 2n)(9m^2 - 6mn + 4n^2)$
   • C. $(3m + 2n)(9m^2 - 6mn + 4n^2)$
   • D. $(3m + 2n)(9m^2 + 6mn + 4n^2)$

3. 3. Factor the sum of cubes $y^3 + 216$ completely. The factored form is $(y+6)(___)$.

4. 4. Which expression is the correct factorization of $p^3 - 1$?

   • A. $(p - 1)(p^2 + p + 1)$
   • B. $(p - 1)(p^2 - p + 1)$
   • C. $(p + 1)(p^2 - p + 1)$
   • D. $(p + 1)(p^2 + p + 1)$

5. 5. When factoring the expression $125x^3 - 64$, the binomial factor is ___.

6. 6. Which expression is the correct factorization of the sum of cubes $y^3 + 64$?

   • A. $(y + 4)(y^2 + 4y + 16)$
   • B. $(y + 4)(y^2 - 4y + 16)$
   • C. $(y - 4)(y^2 + 4y + 16)$
   • D. $(y + 4)(y^2 - 16)$

7. 7. The expression $27x^3 - 1$ is factored into $(3x - 1)(9x^2 + \text{\_\_\_} + 1)$. What is the missing term?

8. 8. A student incorrectly factored $m^3 - 125$ as $(m - 5)(m^2 - 25)$. What is the primary error in this factorization?

   • A. The signs in the quadratic factor are wrong.
   • B. The quadratic factor should be a trinomial, not a binomial.
   • C. The linear factor should be $(m + 5)$.
   • D. The constant term in the quadratic factor is incorrect.

9. 9. When $125p^3 + 8$ is factored into $(5p + 2)(25p^2 + C \cdot p + 4)$, what is the value of the coefficient $C$? ___

10. 10. Which of the following statements correctly shows the factorization of a sum or difference of cubes?

    • A. $x^3 - 27 = (x - 3)(x^2 - 3x + 9)$
    • B. $x^3 + 1 = (x + 1)(x^2 + x + 1)$
    • C. $8x^3 - 1 = (2x - 1)(4x^2 + 2x + 1)$
    • D. $x^3 + 64y^3 = (x + 4y)(x^2 - 4xy + 4y^2)$