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Questions and Answers
What is the root of a polynomial?
What is the root of a polynomial?
A value for the variable of a polynomial that makes the polynomial equal to zero.
To find the roots of a polynomial, it is often useful to find the ____ of the polynomial.
To find the roots of a polynomial, it is often useful to find the ____ of the polynomial.
factors
What is the result of multiplying the matrices |1,2,3| and |3,1,-1|?
What is the result of multiplying the matrices |1,2,3| and |3,1,-1|?
2
Roots of a polynomial are the values that make the polynomial equal to ___.
Roots of a polynomial are the values that make the polynomial equal to ___.
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_____ number roots of a polynomial are the points where the graph of the related polynomial function crosses the x-axis.
_____ number roots of a polynomial are the points where the graph of the related polynomial function crosses the x-axis.
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What is f(g(x)) if f(x)=3x+5 and g(x)=5x-1?
What is f(g(x)) if f(x)=3x+5 and g(x)=5x-1?
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What is the nature of the roots of the equation y=x²+4?
What is the nature of the roots of the equation y=x²+4?
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If abc=0, then ___.
If abc=0, then ___.
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How many real roots are there in y=x³-3x²+x-3 based on the graph description (left down right side up)?
How many real roots are there in y=x³-3x²+x-3 based on the graph description (left down right side up)?
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What are the correct roots of the polynomial equation x³+4x²-11x-30=(x+2)(x-3)(x+5)=0?
What are the correct roots of the polynomial equation x³+4x²-11x-30=(x+2)(x-3)(x+5)=0?
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What are the roots of the polynomial equation (2x+3)(3x-5)=0?
What are the roots of the polynomial equation (2x+3)(3x-5)=0?
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Where does the graph of the polynomial function f(x)=x²+5x+6 cross the x-axis?
Where does the graph of the polynomial function f(x)=x²+5x+6 cross the x-axis?
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Use the quadratic formula to find the roots of the quadratic equation y=x²+x+1.
Use the quadratic formula to find the roots of the quadratic equation y=x²+x+1.
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Study Notes
Polynomial Roots
- Roots of a polynomial are the values that make the polynomial equal to zero.
- Real number roots represent points where the polynomial's graph intersects the x-axis.
Finding Roots
- To find polynomial roots, identifying factors of the polynomial proves useful.
- If the product of factors equals zero (abc=0), then at least one factor must be zero (a=0, b=0, or c=0).
Graph Analysis
- The function y=x²+4 has a minimum value of 4, indicating it has no real roots since it doesn't intersect the x-axis.
- The graph of the polynomial function f(x) = x² + 5x + 6 crosses the x-axis at x = -2 and x = -3.
Matrix Multiplication
- Matrix multiplication for |1,2,3| and |3| results in |2|.
Function Composition
- For functions f(x) = 3x + 5 and g(x) = 5x - 1, the composition f(g(x)) equals 15x + 2.
Root Calculation Examples
- The real roots for y=x³-3x²+x-3 can be obtained by graph inspection, yielding three roots.
- The polynomial equation x³+4x²-11x-30 can be factored into (x+2)(x-3)(x+5), with roots x = -2, 3, and -5.
- The roots of the equation (2x+3)(3x-5)=0 are found to be x = -3/2 and x = 5/3.
Quadratic Formula
- Using the quadratic formula, roots for the equation y=x²+x+1 are x = -½ ± i√3/₂, indicating complex roots.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of polynomial roots and matrices with these flashcards from Algebra 2 Unit 6 Lesson 5. Each card presents key definitions and concepts that are essential for mastering algebraic principles. Perfect for quick revision and retention of important terms!
