Invertible Polynomials: P2 (R) → P2 (R)

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Start date Dec 10, 2020
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Algebra 2 Linear algebra Polynomials

In summary, an invertible polynomial in P2 (R) is a polynomial of degree 2 with real coefficients that has an inverse polynomial in P2 (R). This means that when the polynomial is multiplied by its inverse, the result is the constant polynomial 1. To determine if a polynomial in P2 (R) is invertible, one can use the determinant method. The inverse of an invertible polynomial in P2 (R) is another polynomial that, when multiplied by the original, results in the constant polynomial 1. An invertible polynomial in P2 (R) can only have real coefficients, as the inverse polynomial must also have real coefficients. Invertible polynomials in P2 (R) are useful in

Dec 10, 2020

username123456789

TL;DR Summary: Stuck in this Invertible polynome

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Let T: P2 (R) → P2 (R) be the linear map defined by T(p(x)) = p''(x) - 5p'(x). Is T invertible ? P2 (R) is the vector space of polynomials of degree 2 or less

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Dec 10, 2020

Office_Shredder

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For it to be invertible it has to be both 1 to 1 and onto. To start, you might want to just write a polynomial in terms of its coefficients and see what T actually gives.

1. What is an invertible polynomial?

An invertible polynomial is a polynomial function that has an inverse function. This means that for every input, there is a unique output, and for every output, there is a unique input. In other words, the polynomial can be "reversed" or "undone" by its inverse function.

2. How do you determine if a polynomial is invertible?

A polynomial is invertible if and only if it is a one-to-one function, meaning that it passes the horizontal line test. This can be determined by graphing the polynomial or by checking the values of its coefficients.

3. What is the domain and range of an invertible polynomial?

The domain of an invertible polynomial is the set of all real numbers, denoted as P2 (R). The range is also P2 (R), which represents the set of all possible outputs of the polynomial function.

4. How do you find the inverse of an invertible polynomial?

To find the inverse of an invertible polynomial, you can use the process of polynomial long division. This involves dividing the polynomial by x and then swapping the positions of the coefficients. The resulting polynomial is the inverse function.

5. Can an invertible polynomial have a degree higher than 2?

Yes, an invertible polynomial can have a degree higher than 2. The degree of the polynomial does not affect its invertibility, as long as it is a one-to-one function. However, the notation P2 (R) specifically refers to polynomials of degree 2.