Questions and Answers
What is the completely factored form of the polynomial $x^3 - 8x^2 - 2x + 16$?
What is the completely factored form of the polynomial $3x^5 - 75x^3$?
What is the completely factored form of the polynomial $(x + 3)(x - 2)(2x)(x + 2)$?
What is the completely factored form of the polynomial $x^5 - x^4 + 3x - 3$?
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What is the completely factored form of the polynomial $(3x)(x-4)(x - 1)^2$?
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What is the completely factored form of $7x^5 - 21x^4 + 14x^3$?
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What is the result of $(3x^5 - x)/x$?
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What does $(2x^4 + 4x^3 - x^2)/x$ simplify to?
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What is $(x^4 + 5x^3 + 3x^2)/x$ evaluated?
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What does $(6x^5 - 2x^4 - 1)/x$ equal?
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The value of $c$ so that the polynomial $p(x) = -x^3 + cx^2 - 4x + 3$ is divisible by $(x - 3)$ is ____.
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Select all polynomials that are divisible by $(x - 1)$:
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What is $3!$?
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How many different ways can we arrange 5 different robots?
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What is $200!/199!$?
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Find the value of $c$ so that the polynomial $p(x) = 4x^3 + cx^2 + x + 2$ is divisible by $(x + 2)$.
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Select all polynomials that have $(x + 2)$ as a factor:
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Study Notes
Polynomial Factoring Techniques
- Grouping is effective for factoring expressions with four terms by creating two pairs and finding their greatest common factors (GCF).
- Example: For (x - 8)(x^2 - 2), factoring by grouping yielded (x - 8)(x^2 - 2) after extracting common factors.
Polynomial Definitions
- Complete Factorization: Refers to breaking down a polynomial into its simplest multiplicative form.
- Example: x^3 - 8x^2 - 2x + 16 can be factored to (x - 8)(x^2 - 2).
Higher-Degree Polynomials
- Factor larger polynomial expressions by identifying possible factors and using polynomial division.
- Example: 3x^3 (x + 5)(x - 5) simplifies to 3x^5 - 75x^3.
Special Polynomial Cases
- Expressions like (3x)(x-4)(x - 1)^2 demonstrate factoring into simpler polynomial forms.
- Example: 2x^3 + 4x^2 - x can be expressed as (2x^4 + 4x^3 - x^2)/x.
Practical Applications of Polynomial Division
- Understanding polynomial long division and synthetic division is crucial for simplifying expressions.
- Example: For x + 3, division results in (x^2 + 6x + 9)/(x + 3) which reveals the common factor.
Zero Conditions for Polynomial Divisibility
- Setting polynomials equal to zero at given points helps find values for coefficients ensuring divisibility.
- Example: For p(x) = -x^3 + cx^2 - 4x + 3, finding c for divisibility by (x-3) is done by substituting x = 3.
Factorial Concepts
- Factorials (n!) represent the product of all positive integers up to n.
- Example: 3! = 3 × 2 × 1 = 6 and 5! = 5 × 4 × 3 × 2 × 1 = 120.
Choosing Factors in Polynomial Expressions
- Selecting functions with shared factors involves testing specific values (e.g., x = 1) in equations.
- Example: Identifying polynomials a) A(x), b) B(x) that are divisible by (x - 1) through evaluation at x = 1.
Polynomial Degree Implications
- Degree influences the potential complexity of a polynomial's factors; higher degrees generally indicate multiple factors and roots.
- Example: For p(x) = 4x^3 + cx^2 + x + 2, determining c such that (x + 2) is a factor simplifies calculations.
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Description
Explore factoring by grouping in Algebra II with these flashcards. Learn to identify common factors in polynomial expressions to simplify them effectively. Perfect for students looking to strengthen their algebra skills.
