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**Hint:**In both of the cases we are supposed to consider the number as \[x\]. In the first case, we will be multiplying this variable value with \[4\] and then we subtract \[20\] and equate it to \[360.4\]. Upon solving this, we get the required answer. In the second case, we will be multiplying the number five times and then subtract \[63\] and equate it to half of the original number. And upon solving this equation, we obtain the required answer.

**Complete step-by-step solution:**

Now let us learn about polynomial equations. A polynomial expression with an equal symbol is called a polynomial equation. Polynomial expression with a single term is known as monomial, with two terms is known as binomial, with three terms is known as trinomial, and with 4 or more than 4 terms are known as multinomial. These polynomials are further classified into linear, quadratic, cubic on the basis of the degree of the polynomial.

Now let us consider the first case and solve it.

Let the number be \[x\].

Given that , the number is multiplied by \[4\] and then \[20\] is subtracted from the result, we get \[360.4\].

Expressing this numerically and solving it, we get

\[\begin{align}

& \Rightarrow 4x-20=360.4 \\

& \Rightarrow 4x=380.4 \\

& \Rightarrow x=\dfrac{380.4}{4}=95.1 \\

\end{align}\]

\[\therefore \] The number is \[95.1\].

Now let us consider the second case and solve it.

We are given that , if \[63\] is subtracted from five times a number, the result is half the original number.

Expressing this numerically and solving it, we get

\[\begin{align}

& \Rightarrow 5x-63=\dfrac{x}{2} \\

& \Rightarrow 5x-\dfrac{x}{2}=63 \\

& \Rightarrow \dfrac{10x-x}{2}=63 \\

& \Rightarrow 9x=126 \\

& \Rightarrow x=14 \\

\end{align}\]

**\[\therefore \] The number is \[14\].**

**Note:**We must have a note that the most general rule of a polynomial is that it must contain no square roots of variables, no fractional or negative powers on the variables, and no variables in the denominators of any fractions. The degree of the polynomial is nothing but the highest power of the term in the expression. The value which solves the expression or equation is known as polynomial value

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