Algebra: Perfect Squares and Cubes

Difference of Perfect Squares

• Refers to an expression in the form ( a^2 - b^2 )
• Can be factored using the formula ( (a - b)(a + b) )
• Example: ( x^2 - 9 ) can be factored to ( (x - 3)(x + 3) )

Difference of Perfect Cubes

• Represents an expression in the form ( a^3 - b^3 )
• Factored using the formula ( (a - b)(a^2 + ab + b^2) )
• Example: ( 8 - 27 ) can be factored as ( (2 - 3)(2^2 + 2 \cdot 3 + 3^2) ) which simplifies to ( (2 - 3)(4 + 6 + 9) )

Sum of Perfect Cubes

• Describes an expression in the form ( a^3 + b^3 )
• Can be factored using the formula ( (a + b)(a^2 - ab + b^2) )
• Example: ( 1 + 27 ) can be expressed as ( (1 + 3)(1^2 - 1 \cdot 3 + 3^2) )

Sum of Perfect Squares

• Describes a situation with an expression of the form ( a^2 + b^2 )
• Cannot be factored over the real numbers or integers
• Example: ( 4 + 9 ) cannot be factored further, remaining as ( 13 )

Explore the differences and sums of perfect squares and cubes with this quiz. Learn how to factor expressions such as a² - b², a³ - b³, and a³ + b³. Test your understanding through examples and formulas.