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.

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

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

  2. Given $a^3 - b^3 = 100$ and $a - b = 5$, find the value of $a^2 + b^2$. Round to two decimal places.

    • $21.67
    • $20.00
    • $25.00
    • $-5.00

    Answer: $21.67

  3. If $x^4 - y^4 = 240$ and $x^2 + y^2 = 30$, what is the value of $x^2 - y^2$?

    • $8
    • $7{,}200
    • $-210
    • $5

    Answer: $8

  4. Let $M = a+b$ and $N = a^2 - ab + b^2$. Which of the following expressions represents $a^3 + b^3$?

    • $MN$
    • $M+N$
    • $M^2N$
    • $M/N$

    Answer: $MN$

  5. Factor $x^6 - 64$ completely.

    • $(x - 2)(x + 2)(x^2 + 2x + 4)(x^2 - 2x + 4)$
    • $(x^3 - 8)(x^3 + 8)$
    • $(x^2 - 4)(x^4 + 4x^2 + 16)$
    • $(x - 2)(x^5 + 2x^4 + 4x^3 + 8x^2 + 16x + 32)$

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

  6. Calculate the exact value of $100^3 - 99^3$.

    • $29{,}701
    • $199
    • $19{,}801
    • $1{,}970{,}299

    Answer: $29{,}701

  7. Simplify the expression $\frac{x^4 - y^4}{x^3 + y^3}$, assuming $x \ne -y$.

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

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

  8. Given $a+b=5$ and $a-b=3$, find the value of $a^2 - ab + b^2$. Round to two decimal places.

    • $13.00
    • $17.00
    • $21.00
    • $4.00

    Answer: $13.00

  9. Factor $x^8 - y^8$ completely.

    • $(x - y)(x + y)(x^2 + y^2)(x^4 + y^4)$
    • $(x^4 - y^4)(x^4 + y^4)$
    • $(x^2 - y^2)(x^6 + x^4y^2 + x^2y^4 + y^6)$
    • $(x - y)(x^7 + x^6y + x^5y^2 + x^4y^3 + x^3y^4 + x^2y^5 + xy^6 + y^7)$

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

  10. If $a^3 - b^3 = K$ and $a^3 + b^3 = L$, express $a^6 - b^6$ in terms of $K$ and $L$.

    • $KL$
    • $K+L$
    • $K-L$
    • $\frac{K}{L}$

    Answer: $KL$

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