Questions and Answers
What are the simple zeros of the function $f(x) = x^2 - x - 2$?
2 and -1
What are the zeros with multiplicity for the function $f(x) = (x - 1)^2$?
1 with multiplicity 2
What are the zeros with multiplicity for the function $f(x) = x^3$?
0 with multiplicity 3
How many zeros does any nth order polynomial have?
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For what type of polynomial does at least one real zero always exist?
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Study Notes
Root Finding Problems
- Root finding problems involve finding the roots of equations, which is a common requirement in many problems.
- A root finding problem is defined as finding the value of r such that f(r) = 0, where f(x) is a continuous function.
Roots of Equations
- A root of an equation is a number that satisfies the equation.
- Example: The equation x^2 + 2x - 3 = 0 has 3 roots: two simple roots (-1 and -2) and one repeated root (3) with a multiplicity of 2.
Zeros of a Function
- A zero of a function f(x) is a number r for which f(r) = 0.
- Example: The function f(x) = x^2 - 2(x - 3) has 2 zeros: 2 and 3.
Graphic Interpretation of Zeros
- The real zeros of a function f(x) are the values of x at which the graph of the function crosses or touches the x-axis.
Types of Solutions for Non-Linear Equations
- There are two types of solutions for non-linear equations: bracketing and open methods.
Numerical Methods for Solving Non-Linear Equations
- Bisection Method
- Newton Raphson Method
- The Secant Method
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