Podcast
Questions and Answers
The numerators of any rational roots of a polynomial will be the factors of the ___ term.
The numerators of any rational roots of a polynomial will be the factors of the ___ term.
- Linear
- Quadratic
- Leading
- Constant (correct)
What is the product of (2x + 3)(2x^2 - 6x + 7)?
What is the product of (2x + 3)(2x^2 - 6x + 7)?
4x^3 - 6x^2 - 4x + 21
Each number in Pascal's Triangle is the ___ of the two numbers directly above it in the previous row.
Each number in Pascal's Triangle is the ___ of the two numbers directly above it in the previous row.
- Difference
- Quotient
- Product
- Sum (correct)
The graph of the ___ of a polynomial function is the reflection of the graph of the polynomial function over the line y=x.
The graph of the ___ of a polynomial function is the reflection of the graph of the polynomial function over the line y=x.
What are the factors of the constant?
What are the factors of the constant?
What are the factors of the leading coefficient?
What are the factors of the leading coefficient?
What are the roots?
What are the roots?
Find the inverse of the function y=10x^2 - 4.
Find the inverse of the function y=10x^2 - 4.
What is an estimate for the domain and range of the graph of the polynomial function?
What is an estimate for the domain and range of the graph of the polynomial function?
Find the composition of the function f(g(x)) where f(x)=4x^2 and g(x)=2x/4.
Find the composition of the function f(g(x)) where f(x)=4x^2 and g(x)=2x/4.
Factor the polynomial 125x^3.
Factor the polynomial 125x^3.
Use Pascal's Triangle to help find the missing values for x^4 - 4x^3 + 6x^2 - ax + 1. What is a?
Use Pascal's Triangle to help find the missing values for x^4 - 4x^3 + 6x^2 - ax + 1. What is a?
What is the domain of a polynomial function?
What is the domain of a polynomial function?
Is the inverse of the function y=x^5 a function?
Is the inverse of the function y=x^5 a function?
Find the composition of the function g(f(x)) where f(x)=2x and g(x)=x^2 + 2x - 1.
Find the composition of the function g(f(x)) where f(x)=2x and g(x)=x^2 + 2x - 1.
The denominators of any rational roots of a polynomial will be the factors of the ___ coefficient.
The denominators of any rational roots of a polynomial will be the factors of the ___ coefficient.
A shortcut method of dividing a polynomial by a linear polynomial using only the coefficients is called ___.
A shortcut method of dividing a polynomial by a linear polynomial using only the coefficients is called ___.
Divide using synthetic division: -3 | 1 8 9 -18.
Divide using synthetic division: -3 | 1 8 9 -18.
Divide (6x^3 + 26x^2 + 16x - 24) by (x + 3).
Divide (6x^3 + 26x^2 + 16x - 24) by (x + 3).
If the highest exponent of a polynomial function is ___, then the range of the function is always all real numbers.
If the highest exponent of a polynomial function is ___, then the range of the function is always all real numbers.
Factor the polynomial 216x^3 - 64.
Factor the polynomial 216x^3 - 64.
Divide using synthetic division: 5 | 8 -38 -16 30.
Divide using synthetic division: 5 | 8 -38 -16 30.
Flashcards
Rational Root Numerators
Rational Root Numerators
Factors of the constant term.
Rational Root Denominators
Rational Root Denominators
Factors of the leading coefficient.
Multiplying Polynomials
Multiplying Polynomials
Applying distributive property across terms.
Function Composition
Function Composition
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Pascal's Triangle Formation
Pascal's Triangle Formation
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Graph of an Inverse Function
Graph of an Inverse Function
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Domain of Polynomial Functions
Domain of Polynomial Functions
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Range of Polynomial Functions
Range of Polynomial Functions
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Synthetic Division
Synthetic Division
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Function Composition
Function Composition
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Factoring Polynomials
Factoring Polynomials
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Essential Polynomial Operations
Essential Polynomial Operations
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Inverse Function Graphs
Inverse Function Graphs
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Study Notes
Rational Roots and Polynomials
- The numerators of rational roots are factors of the constant term.
- The denominators of rational roots are factors of the leading coefficient.
Polynomial Operations
- To multiply polynomials, use distributive properties: ( (2x + 3)(2x^2 - 6x + 7) = 4x^3 - 6x^2 - 4x + 21 ).
- Finding the composition of functions involves substituting one function into another, e.g., ( f(g(x)) ) results in ( x^2 ) for ( f(x) = 4x^2 ) and ( g(x) = \frac{2x}{4} ).
Pascal's Triangle
- Each number in Pascal's Triangle is the sum of the two numbers directly above it, aiding in combinatorial calculations.
Inverses of Functions
- The graph of the inverse of a polynomial function reflects over the line ( y = x ).
- Example of finding an inverse: ( y = 10x^2 - 4 ) gives ( y = \frac{x + 4}{10} ).
Roots and Factors
- The rational roots can include: 1, ( \frac{1}{5} ), 7, ( \frac{7}{5} ).
- Factors of a polynomial can be derived, for instance, ( 125x^3 ) factors into ( (5x + 3)(25x^2 - 15x + 9) ).
Domain and Range
- The domain of a polynomial function is always all real numbers.
- The range can vary depending on the highest exponent of the polynomial; if it is odd, the range is all real numbers.
Polynomial Division
- Synthetic Division is a shortcut method for dividing polynomials using only coefficients.
- Example of synthetic division with ( -3 ): ( 1 , 8 , 9 , -18 ) results in ( x^2 + 5x - 6 ).
Function Composition
- Composing functions like ( g(f(x)) ) for ( f(x) = 2x ) and ( g(x) = x^2 + 2x - 1 ) yields ( 4x^2 + 4x - 1 ).
Factoring Polynomials
- Example of factoring: ( 216x^3 - 64 ) factors to ( (6x - 4)(36x^2 + 24x + 16) ).
- Utilizing Pascal's Triangle assists in identifying coefficients in polynomial expansions.
Key Skills
- Understand the structure and properties of polynomial functions.
- Master operations including multiplication, division (synthetic and long), and factorization.
- CAPTIVATING the inverse relationship between functions and their graphs through reflection principles.
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