Factoring Formulas — Hard Practice Quiz

A Algebra cheat sheet for Factoring Formulas — every key formula with its symbols defined — plus a hard-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 $x^6 - y^6$ completely.

    • $(x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)$
    • $(x^2 - y^2)(x^4 + x^2y^2 + y^4)$
    • $(x^3 - y^3)(x^3 + y^3)$
    • $(x - y)(x + y)(x^4 + x^2y^2 + y^4)$

    Answer: $(x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)$

  2. Simplify the expression $\frac{x^6 - y^6}{x^3 - y^3}$.

    • $x^3 + y^3$
    • $(x + y)(x^2 - xy + y^2)$
    • $x^2 + xy + y^2$
    • $x^2 - y^2$

    Answer: $(x + y)(x^2 - xy + y^2)$

  3. If $x^3 + y^3 = 10$ and $x + y = 2$, find the value of $xy$.

    • $1$
    • $-1$
    • $\frac{1}{3}$
    • $-\frac{1}{3}$

    Answer: $-\frac{1}{3}$

  4. Which of the following is NOT a factor of $x^8 - y^8$?

    • $x^2 + y^2$
    • $x^4 + y^4$
    • $x^3 + y^3$
    • $x - y$

    Answer: $x^3 + y^3$

  5. If $A = x^2 - y^2$ and $B = x^3 + y^3$, what is $\frac{B}{A}$ in simplest form, assuming $x \ne y$ and $x \ne -y$?

    • $\frac{x^2 + xy + y^2}{x + y}$
    • $\frac{x^2 - xy + y^2}{x - y}$
    • $x - y$
    • $x + y$

    Answer: $\frac{x^2 - xy + y^2}{x - y}$

  6. If $a - b = 3$ and $a^3 - b^3 = 63$, find the value of $a^2 + b^2$.

    • $17$
    • $13$
    • $21$
    • $9$

    Answer: $17$

  7. Factor $x^7 - xy^6$ completely.

    • $x(x^3 - y^3)(x^3 + y^3)$
    • $x(x - y)(x + y)(x^4 + x^2y^2 + y^4)$
    • $x(x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)$
    • $x(x^2 - y^2)(x^4 + x^2y^2 + y^4)$

    Answer: $x(x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)$

  8. If $x^2 + y^2 = 10$ and $xy = 3$, what is the value of $x^4 - y^4$?

    • $80$
    • $-80$
    • $64$
    • $100$

    Answer: $80$

  9. Simplify the expression $\frac{x^6 - y^6}{(x^2 - y^2)(x^3 + y^3)}$.

    • $\frac{x^2 - xy + y^2}{x - y}$
    • $\frac{x^2 + xy + y^2}{x + y}$
    • $x + y$
    • $x - y$

    Answer: $\frac{x^2 + xy + y^2}{x + y}$

  10. If $a^2 + b^2 = 10$ and $ab = 3$, what is the value of $|a^5 - b^5|$?

    • $242$
    • $121$
    • $100$
    • $82$

    Answer: $242$

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