Factoring Formulas — Practice Quiz

A Algebra cheat sheet for Factoring Formulas — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Difference of Squares

$$a^2 - b^2 = (a - b)(a + b)$$

Difference of Cubes

$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

Sum of Cubes

$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$

Difference of Fourth Powers

$$a^4 - b^4 = (a - b)(a + b)(a^2 + b^2)$$

Difference of Fifth Powers

$$a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$$

Practice quiz

  1. Factor the expression $25x^2 - 49y^2$.

    • $(5x - 7y)(5x + 7y)$
    • $(5x - 7y)^2$
    • $(25x - 49y)(25x + 49y)$
    • $(5x + 7y)^2$

    Answer: $(5x - 7y)(5x + 7y)$

  2. Which of the following is the correct factorization of $8a^3 + 125b^3$?

    • $(2a + 5b)(4a^2 - 10ab + 25b^2)$
    • $(2a + 5b)(4a^2 + 10ab + 25b^2)$
    • $(2a - 5b)(4a^2 + 10ab + 25b^2)$
    • $(2a + 5b)^3$

    Answer: $(2a + 5b)(4a^2 - 10ab + 25b^2)$

  3. Factor the expression $x^3 - 64$.

    • $(x - 4)(x^2 + 4x + 16)$
    • $(x + 4)(x^2 - 4x + 16)$
    • $(x - 4)(x^2 - 4x + 16)$
    • $(x - 4)^3$

    Answer: $(x - 4)(x^2 + 4x + 16)$

  4. What is the complete factorization of $16m^4 - n^4$?

    • $(2m - n)(2m + n)(4m^2 + n^2)$
    • $(4m^2 - n^2)(4m^2 + n^2)$
    • $(2m - n)(2m + n)(2m - n)(2m + n)$
    • $(4m^2 - n^2)^2$

    Answer: $(2m - n)(2m + n)(4m^2 + n^2)$

  5. Factor the expression $x^5 - 32$.

    • $(x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)$
    • $(x - 2)(x^4 - 2x^3 + 4x^2 - 8x + 16)$
    • $(x + 2)(x^4 - 2x^3 + 4x^2 - 8x + 16)$
    • $(x - 2)^5$

    Answer: $(x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)$

  6. Which expression is equivalent to $(x^2 - y^2)(x^2 + y^2)$?

    • $x^4 - y^4$
    • $x^4 + y^4$
    • $(x - y)^2(x + y)^2$
    • $x^4 - 2x^2y^2 + y^4$

    Answer: $x^4 - y^4$

  7. What is the complete factorization of $a^6 - b^6$?

    • $(a - b)(a + b)(a^2 + ab + b^2)(a^2 - ab + b^2)$
    • $(a^2 - b^2)(a^4 + a^2b^2 + b^4)$
    • $(a^3 - b^3)^2$
    • $(a - b)^6$

    Answer: $(a - b)(a + b)(a^2 + ab + b^2)(a^2 - ab + b^2)$

  8. If $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$, what is the value of $10^3 + 2^3$?

    • $1008$
    • $1012$
    • $1024$
    • $1000$

    Answer: $1008$

  9. Which of the following is NOT a factor of $x^4 - 1$?

    • $(x - 1)$
    • $(x + 1)$
    • $(x^2 + 1)$
    • $(x^2 + x + 1)$

    Answer: $(x^2 + x + 1)$

  10. If $a - b = 3$ and $a^2 + ab + b^2 = 7$, what is the value of $a^3 - b^3$?

    • $10$
    • $21$
    • $28$
    • $30$

    Answer: $21$

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