Factoring Quiz for Algebra 2 Students

Study Notes

2(x - 5)(x + 5) translates to 2x² - 50, demonstrating the difference of squares.
(2x - 1)(4x² + 2x + 1) results in 8x³ - 1, showcasing the difference of cubes.
5(x - 2)(x + 2) simplifies to 5x² - 20, reinforcing the difference of squares.
(xy + 3)(x²y² - 3xy + 9) expands into x³y³ + 27, illustrating the sum of cubes.
(3x - 1)(2y + 1) gives 6xy + 3x - 2y - 1, applying the distributive property effectively.

(2x + 3)(4x + y) combines to create 8x² + 2xy + 12x + 3y, showcasing the product of two binomials.
4(x + 3)(x - 3) results in 4x² - 36, highlighting the structure of difference of squares.
7(x + 2)(x² - 2x + 4) leads to 7x³ + 56, exemplifying polynomial expansion.
(x² + 4)(x + 2)(x - 2) simplifies to x⁴ - 16, further illustrating the difference of squares concept.
5(x² - 3) expands to 5x² - 15, emphasizing the distributive property in action.

x²(y² - 5)(y⁴ + 5y² + 25) produces x²y⁶ - 125x², demonstrating advanced polynomial multiplication with a focus on cubes.
xy(x - 1)(x² + x + 1) simplifies to x⁴y - xy, showcasing the association of polynomial decomposition.
3x²(5 + 2y²) results in 15x² + 6x²y², exemplifying the application of the distributive property with multiple variables.

(4x + 3)(7y - 5) expands to 28xy - 20x + 21y - 15, reflecting a complex interaction of terms.
(3x - 1)(9x² + 3x + 1) simplifies to 27x³ - 1, demonstrating the characteristics of polynomial multiplication.
The expression x² + 16 is recognized as a prime polynomial, which cannot be factored further over the real numbers.

Factoring polynomials often involves recognizing patterns such as the difference of squares, sums and differences of cubes, and the distributive property.
Products of polynomials can expand into larger degree polynomials, emphasizing the importance of careful arrangement and combination.
Understanding the characteristics of prime polynomials is essential for advanced factoring techniques.