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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 ___.
_____ 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.
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?
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?
If abc=0, then ___.
If abc=0, then ___.
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)?
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?
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?
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?
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.
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