Best solutions to - "Does every quadratic polynomial defined in real numbers..."

High School
Algebra
Algebra I
Quadratic function and equation

Swebaceacichegh

Answered question

2022-01-29

Does every quadratic polynomial defined in real numbers always define a function?
For example, let a,b,c

\in R

and define

P (x) = a x^{2} + b x + c

Answer & Explanation

Ronald Alvarez

Beginner2022-01-30Added 11 answers

To answer your question in the comments concerning Polynomials on matrices: A Polynomial P can always be defined if you have multiplication and addition in your structure, i.e. matrices can be multiplied and added together. Take for example

P (x) := 2 x^{2} + x

where

P : R^{2 \times 2} \to R^{2 \times 2}

. Then, you can choose any

x \in R^{2 \times 2}

to evaluate P at - for example, choose

x = (\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix})

. Then

\begin{aligned} P ((\begin{array}{c} 1 & 2 \\ 0 & 1 \end{array})) & = 2 {(\begin{array}{c} 1 & 2 \\ 0 & 1 \end{array})}^{2} + (\begin{array}{c} 1 & 2 \\ 0 & 1 \end{array}) \\ = (\begin{array}{c} 1 & 4 \\ 0 & 1 \end{array}) + (\begin{array}{c} 1 & 2 \\ 0 & 1 \end{array}) \\ = (\begin{array}{c} 2 & 6 \\ 0 & 2 \end{array}) \end{aligned}

As you can see, a Polynomial is just a function that looks a specific way (variable with exponent multiplied with coefficient and added together) and can be evaluated at a specific x.

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