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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 $x^3 - 8x^2 - 2x + 16$?
- $(x - 4)(x^3 + 4)$
- $(x^2 - 16)(x + 2)$
- $(x + 8)(x^2 + 2)$
- $(x - 8)(x^2 - 2)$ (correct)
What is the completely factored form of the polynomial $3x^5 - 75x^3$?
What is the completely factored form of the polynomial $3x^5 - 75x^3$?
- $3x^3(x - 5)(x + 5)$ (correct)
- $x^3(3x^2 - 75)$
- $(x^2 - 5)(3x^3)$
- $3x(x^2 - 25)$
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 $(x + 3)(x - 2)(2x)(x + 2)$?
- $(x^2 + 10)(x + 1)$
- $(2x^2 - 6)(x - 4)$
- $(x^2 + x - 6)(2x^2 + 4x)$ (correct)
- It cannot be factored.
What is the completely factored form of the polynomial $x^5 - x^4 + 3x - 3$?
What is the completely factored form of the polynomial $x^5 - x^4 + 3x - 3$?
What is the completely factored form of the polynomial $(3x)(x-4)(x - 1)^2$?
What is the completely factored form of the polynomial $(3x)(x-4)(x - 1)^2$?
What is the completely factored form of $7x^5 - 21x^4 + 14x^3$?
What is the completely factored form of $7x^5 - 21x^4 + 14x^3$?
What is the result of $(3x^5 - x)/x$?
What is the result of $(3x^5 - x)/x$?
What does $(2x^4 + 4x^3 - x^2)/x$ simplify to?
What does $(2x^4 + 4x^3 - x^2)/x$ simplify to?
What is $(x^4 + 5x^3 + 3x^2)/x$ evaluated?
What is $(x^4 + 5x^3 + 3x^2)/x$ evaluated?
What does $(6x^5 - 2x^4 - 1)/x$ equal?
What does $(6x^5 - 2x^4 - 1)/x$ equal?
The value of $c$ so that the polynomial $p(x) = -x^3 + cx^2 - 4x + 3$ is divisible by $(x - 3)$ is ____.
The value of $c$ so that the polynomial $p(x) = -x^3 + cx^2 - 4x + 3$ is divisible by $(x - 3)$ is ____.
Select all polynomials that are divisible by $(x - 1)$:
Select all polynomials that are divisible by $(x - 1)$:
What is $3!$?
What is $3!$?
How many different ways can we arrange 5 different robots?
How many different ways can we arrange 5 different robots?
What is $200!/199!$?
What is $200!/199!$?
Find the value of $c$ so that the polynomial $p(x) = 4x^3 + cx^2 + x + 2$ is divisible by $(x + 2)$.
Find the value of $c$ so that the polynomial $p(x) = 4x^3 + cx^2 + x + 2$ is divisible by $(x + 2)$.
Select all polynomials that have $(x + 2)$ as a factor:
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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