Questions and Answers
What is the expanded form of 2(x - 5)(x + 5)?
What is the expanded form of (2x - 1)(4x² + 2x + 1)?
What is the expanded form of 5(x - 2)(x + 2)?
What is the expanded form of (xy + 3)(x²y² - 3xy + 9)?
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What is the expanded form of (3x - 1)(2y + 1)?
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What is the expanded form of (2x + 3)(4x + y)?
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What is the expanded form of 4(x + 3)(x - 3)?
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What is the expanded form of 7(x + 2)(x² - 2x + 4)?
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What is the expanded form of (x² + 4)(x + 2)(x - 2)?
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What is the expanded form of 5(x² - 3)?
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What is the expanded form of x²(y² - 5)(y⁴ + 5y² + 25)?
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What is the expanded form of xy(x - 1)(x² + x + 1)?
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What is the expanded form of 3x²(5 + 2y²)?
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What is the expanded form of (4x + 3)(7y - 5)?
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What is the expanded form of (3x - 1)(9x² + 3x + 1)?
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What does 'prime' refer to in algebra?
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Study Notes
Factoring Techniques and Outcomes
- 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.
Polynomial Products
- (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.
Higher Degree Polynomials
- 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.
Complex Polynomial Products
- (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.
Summary of Key Concepts
- 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.
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Test your skills with these flashcards on factoring expressions in Algebra 2. Each card presents a polynomial expression alongside its factored form. Perfect for mastering the concept of factoring and preparing for exams.