Perhaps the best thing to do before giving examples of the correct way to carry out the Polynomial Decomposition of a decimal number is to revise the definition of this operation in order to be able to understand each of the exercises presented in its proper mathematical context.
Polynomial decomposition of a decimal number
In this sense, it can be said that Mathematics has generally defined the Polynomial Decomposition of a decimal number as the operation, whose main purpose is to achieve the expression of a number of this nature in the different monomials, which can produce each of their positional values, as well as in the polynomial that will arise from them, once they are related through the sum operation that must be established between them.
Steps to perform the polynomial decomposition of a decimal number
Likewise, Mathematics has indicated the procedure to be followed in case you wish to carry out this operation, and that must be followed progressively, in order to achieve an expression that effectively corresponds to the original decimal number. Then, each one of them:
- Once the decimal number on which the polynomial decomposition must be made has been given, each one of its parts will be taken, that is to say, the whole part (Units) and the decimal part (Incomplete Units) and decomposed in each one of the numbers that should be added, to give as result the number from which it was divided, trying that these numbers take into account each one of the positional values by which the original number is formed.
- When having each one of these numbers, whose total sum makes up again to the decimal number, the notation of each one of them must be obtained, that is to say, each number of these must be abbreviated by means of a power of base 10. It is necessary to remember that this power will have a positive exponent, if the abbreviated number is integer, and a negative exponent if on the contrary it is decimal.
- Once each number in which the original number has been decomposed has managed to find its scientific notation, and these are related by means of the sum operation, the polynomial decomposition is considered to have been carried out, since each element has been expressed as a monomial (ax3) in which the incognita has been cleared, and these in turn as a polynomial (ax3 + bx2).
- If you want to verify the operation, it will be enough to convert each one of the scientific notations to the numbers that were abbreviated, and add these. The result must be the decimal number that gave rise to the Polynomial Decomposition exercise.
Examples of how to perform Polynomial Decomposition
However, the most efficient way to complete an explanation about this operation, and the correct way in which it must be solved, may be through the exposition of a series of examples that allow us to see in practice how each one of the steps, indicated by Mathematics, should be executed. Next, the following exercises:
Perform polynomial decomposition of the following number: 345
The first example discussed will not reflect how to perform polynomial decomposition in a decimal number, but in an integer, in order to see how to proceed later on as to the whole parts of the decimal numbers that must be submitted to this operation. Consequently, we will begin by decomposing this number in the quantity that each one of its positional values should represent, in order to give as result, in case of adding, the original number that has been raised:
345 → 300 + 40 + 5
Once you have each of these quantities, what should be done is the conversion to its expression as scientific notation. Taking into account that they are integers, all the exponents to which the powers of 10 are elevated by which each one of these numbers will be multiplied must then be positive:
300 + 40 + 5 = 3 . 102 + 4. 101 + 5 . 100
Once this is done, the operation will be considered concluded, so it will be enough to express its result:
345 = 3 . 102 + 4. 101 + 5 . 100
Perform polynomial decomposition of the following number: 0.65
If the number to be decomposed is a decimal number with an integer equal to 0, then simply take into account the numbers that make up the incomplete units of the figure originally stated. Consequently, the decimal part of this number must simply be decomposed, taking into account the different positional values, and the decimal numbers that would in turn make up this number:
0,65 → 0,6 + 0,05
Once these numbers have been found, their expression must also be determined as a corresponding scientific notation in order to achieve polynomial decomposition:
0,6 + 0,05 = 6 . 10-1 + 5 . 10-2
It is then assumed that the operation has been completed, so the result is expressed:
0,65 = 6 . 10-1 + 5 . 10-2
Perform polynomial decomposition of the following number: 3845,987
In this case, it will be a decimal number, which has a whole part other than zero, as well as a decimal part. To begin with the decomposition, it will be necessary to decompose each one of these parts in the numbers, that taking into account the positional value of each one of them, when adding up, would compose the number:
3845 → 3000 + 800 + 40 + 5
987 → 0,9 + 0,08 + 0,007
Once this decomposition has been achieved, then each of the figures obtained should be expressed as Scientific Notation:
3000 + 800 + 40 + 5 → 3 . 103 + 8 . 102 + 4 . 101 + 5 . 100
0,9 + 0,08 + 0,007 → 9 . 10-1 + 8 . 10-2 + 7 . 10-3
Once these abbreviations have been obtained, each of the monomials that originated in the process will be arranged in order, which will then be considered the final solution of the operation:
3845,987 → 3 . 103 + 8 . 102 + 4 . 101 + 5 . 100 + 9 . 10-1 + 8 . 10-2 + 7 . 10-3