Are all polynomials rational functions? Are all algebraic functions polynomials?

Step 1. Are all the functions of polynomials rational?
A polynomial function is defined as a function that can be expressed as a finite sum of terms, each consisting of a variable raised to a non-negative integer power, multiplied by a constant coefficient. The coefficients and exponents can be any rational numbers.
A rational function, on the other hand, is a function that can be expressed as the quotient of two polynomials, where the denominator polynomial is not equal to zero.
Not all functions of polynomials are rational. Polynomial functions can have irrational coefficients or exponents, which would make the function itself irrational. For example, the function $f(x)=\sqrt{2}{x}^2+\pi x+1$ is a polynomial function, but it is not a rational function since it contains irrational coefficients ($\sqrt{2}$ and $\pi$).
Therefore, we can conclude that not all functions of polynomials are rational.
Step 2. Are all polynomials in algebraic equations?
An algebraic equation is an equation that involves one or more polynomials. Polynomials are commonly used in algebraic equations to represent mathematical relationships between variables.
However, not all polynomials are necessarily present in algebraic equations. Some equations may involve other types of mathematical expressions, such as exponential functions, trigonometric functions, or logarithmic functions. In such cases, polynomials may not be present.
For example, the equation $\sin (x)+e^x=2x$ involves a trigonometric function ($\sin (x)$), an exponential function ($e^x$), and a linear function ($2x$). Although there is no explicit polynomial in this equation, it still falls under the category of algebraic equations.
In conclusion, while polynomials are commonly used in algebraic equations, not all equations exclusively involve polynomials.

The answer to the first question, ''Are all the functions of polynomials rational?'' is no. Not all functions of polynomials are rational.
As for the second question, ''Are all polynomials in algebraic equations?'' the answer is yes. All polynomials can be used in algebraic equations.

1. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division by variables. A rational function, on the other hand, is a function defined as the quotient of two polynomials, where the denominator polynomial is not zero.
Let's consider a polynomial function $f(x)$ and analyze its rationality. If all the coefficients of $f(x)$ are rational numbers, then $f(x)$ is a rational function. However, it is possible for a polynomial function to have irrational coefficients, in which case it would not be a rational function.
For example, the polynomial $f(x)=x^2-\sqrt{2}$ has an irrational coefficient ($-\sqrt{2}$). Hence, $f(x)$ is not a rational function.
In summary, not all functions of polynomials are rational. Some polynomial functions may have irrational coefficients, making them irrational functions.
2. An algebraic equation is an equation that involves one or more variables and consists of polynomial expressions. Polynomial equations, which are a type of algebraic equations, are equations in which two polynomials are set equal to each other.
In general, polynomial equations can involve variables, constants, and polynomial expressions. Therefore, all polynomials can be part of algebraic equations.
For example, the equation $x^2+3x-2=0$ is a polynomial equation where the variable $x$ is raised to the power of 2, and the coefficients 1, 3, and -2 are all constants.
In summary, all polynomials can be used in algebraic equations since polynomial equations involve polynomial expressions and variables.