Questions and Answers
The numerators of any rational roots of a polynomial will be the factors of the ___ term.
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.
The graph of the ___ of a polynomial function is the reflection of the graph of the polynomial function over the line y=x.
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What are the factors of the constant?
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What are the factors of the leading coefficient?
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What are the roots?
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Find the inverse of the function y=10x^2 - 4.
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What is an estimate for the domain and range of the graph of the polynomial function?
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Find the composition of the function f(g(x)) where f(x)=4x^2 and g(x)=2x/4.
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Factor the polynomial 125x^3.
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Use Pascal's Triangle to help find the missing values for x^4 - 4x^3 + 6x^2 - ax + 1. What is a?
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What is the domain of a polynomial function?
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Is the inverse of the function y=x^5 a function?
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Find the composition of the function g(f(x)) where f(x)=2x and g(x)=x^2 + 2x - 1.
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The denominators of any rational roots of a polynomial will be the factors of the ___ coefficient.
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A shortcut method of dividing a polynomial by a linear polynomial using only the coefficients is called ___.
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Divide using synthetic division: -3 | 1 8 9 -18.
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Divide (6x^3 + 26x^2 + 16x - 24) by (x + 3).
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If the highest exponent of a polynomial function is ___, then the range of the function is always all real numbers.
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Factor the polynomial 216x^3 - 64.
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Divide using synthetic division: 5 | 8 -38 -16 30.
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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.
Studying That Suits You
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Description
Prepare for your Algebra 2 Unit 6 test with these flashcards designed to reinforce key concepts and problem-solving skills. Each card focuses on essential topics such as rational roots, polynomial multiplication, and Pascal's Triangle. Test your knowledge and ace your upcoming exam!
