Special Factorizations

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Difference of Squares

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Theorem:
An expression of the form \( a^2 - b^2 \) can be factored into
\( (a + b)(a - b) \).

Proof:

\[ \begin{aligned} (a + b)(a - b) &= a^2 + ab - ab - b^2 \\ &= a^2 - b^2 \end{aligned} \]

Exercise 1: Calculate the following without a calculator: \( 2024^2 - 2023^2 \).
Exercise 2: Calculate the following without a calculator: \( \frac{2024^2 - 1012^2}{1012} \).

Sum of Cubes

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Theorem:
An expression of the form \( a^3 + b^3 \) can be factored into
\( (a + b)(a^2 - ab + b^2) \).

Exercise 3: Factor the following completely: \( x^3 + 1 \).
Exercise 4: Express in simplest form:
\( \frac{9x^2 - 4y^2}{27x^3 + 8y^3} \)

Difference of Cubes

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Theorem:
An expression of the form \( a^3 - b^3 \) can be factored into
\( (a - b)(a^2 + ab + b^2) \).

Exercise 5: Factor the following completely: \( x^3 - 1 \).
Exercise 6: If \( a \) and \( b \) have a difference of 8, and the difference of their cubes is 32, what is the value of \( a^2 + ab + b^2 \)?

Review

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Exercise 7: Factor completely: \( (8x^3 - 27y^3)(8x^3 + 27y^3) \)
Exercise 8: Factor completely: \( 32x^6 - 729 \)
Exercise 9: If \( a + b = 7 \) and \( a^3 + b^3 = 42 \), what is the value of the sum \( \frac{1}{a} + \frac{1}{b} \)?
(Source: 2004 MATHCOUNTS National Round)
Exercise 10: If \( a \) and \( b \) are real numbers such that \( a + b = 1 \) and \( a^2 + b^2 = 2 \), what is the value of \( a^3 + b^3 \)?
(Source: 2013 MATHCOUNTS National Round)
Exercise 11: What is the value of

\[ \frac{2013^3 - 2 \cdot 2013^2 \cdot 2014 - 3 \cdot 2013 \cdot 2014^2 - 2014^3 + 1}{2013 \cdot 2014} \]

(Source: 2014 MATHCOUNTS National Round)

Exercise 12: If \( (a - \frac{1}{a})^2 = 4 \), what is the absolute value of \( a^3 - \frac{1}{a^3} \)?
(Source: 2016 MATHCOUNTS National Round)

Credits

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All explanations and problems written by Adrian Hernandez Vega, unless otherwise noted.

Prefer video? Watch the full explanation here: