Roots of Polynomials: Understanding Mathematical Methods

In summary, in chapter 1 of the book "Mathematical Methods for Physics and Engineering", it is discussed that the function F(x) can be written as a product of factors (x-α1)(x-α2)...(x-αr). However, it is also mentioned that the same conditions can be met by replacing this with a new function F(x) = A(x-α1)^m1(x-α2)^m2...(x-αr)^mr, where each factor is raised to a power. This may be confusing at first, but it is important to note that the two functions still have the same zeros, meaning that they both equal 0 when x is equal to any of the values
  • #1
Rishabh Narula
61
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I was reading this book - " mathematical methods for physics and engineering"
in it in chapter 1 its says
"F(x) = A(x - α1)(x - α2) · · · (x - αr),"
this makes sense to me but then it also said

We next note that the condition f(αk) = 0 for k = 1, 2, . . . , r, could also be met
if (1.8) were replaced by
F(x) = A(x - α1)^m1(x - α2)^m2 · · · (x - αr)^mr

this confused me...how can you can you just raise each factor to powers m1,m2,...mr etc.How does the function still remain same?
 
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  • #2
The function doesn't "remain the same" and this doesn't say it does. It just says that the two functions have the same zeros. Both [tex](x- a_r)[/tex] and [tex](x- a_r)^n[/tex] are 0 if and only if [tex]x= a_r[/tex].
 
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1. What are the roots of a polynomial?

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation formed by setting the polynomial equal to zero.

2. How do you find the roots of a polynomial?

There are several methods for finding the roots of a polynomial, including factoring, the quadratic formula, and using the rational root theorem. The method used depends on the degree and complexity of the polynomial.

3. Why is it important to understand the roots of polynomials?

Understanding the roots of polynomials is important because they provide valuable information about the behavior of the polynomial, such as the number of real solutions and the shape of the graph. They also have many practical applications in fields such as engineering, physics, and economics.

4. What is the relationship between the roots of a polynomial and its factors?

The roots of a polynomial are the values that make the polynomial equal to zero, while the factors of a polynomial are the expressions that divide evenly into the polynomial. The roots and factors are closely related, as the roots of a polynomial are also the solutions to the equation formed by setting the polynomial equal to zero, which are also the values that make the factors equal to zero.

5. Can a polynomial have complex roots?

Yes, a polynomial can have complex roots. Complex roots are solutions that involve imaginary numbers, which are numbers that involve the square root of -1. Complex roots occur when the polynomial has no real solutions, and they are important in understanding the behavior of the polynomial.

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