Definition of Polynomials

polynomials in x degree n are a form 
anxn + an-1 xn-1 + an-2 xn-2+ … + + a2 x2 + a1x + a0
for n a chunk number, and a0, a1, …, aas a constant and an ≠ 0.

In a polynomial where a1, …, an  is called the tribal coefficient x. The number n is called the polynomial quality. The form of writing a polynomial is arranged with the highest rank base placed in the front row, while the smaller rank is on the right. An example is writing the following polynomial 4.

2x4 – 7x3 + 5x – 9

Description: The coefficient of x is 2, the coefficient of x3 is -7, the coefficient of x is 5, the constant is -9.

Polynomial Value

To facilitate the method of mentioning polynomials it is often expressed by the form function f(x). To determine the value of a polynomial is to replace x with a constant.

Example 1.

Determine the value of the polynomial, 4x4 – 3x + 6 for x = 2
Answer:
f(x) = 4x4 – 3x + 6
substitute the value x = 2 to f(x)
f(2) = 4 . 24 – 3 . 2 + 6 = 64 – 6 + 6 = 64

Such a method is indeed quite simple, but for polynomials with a high enough taste or a relatively high number of tribes, then this method can obviously cause difficulties in the calculation. For this reason, consider a more systematic way to calculate polynomial values ​​as follows,

For example to obtain a polynomial value, consider the following polynomial,
2x3 + 4x2 – 3x + 2

We can express the polynomial with the following form of writing
f(x) = 2x3 + 4x2 – 3x + 2
f(x) = (2x2 + 4x – 3)x + 2

f(x) = [(2x + 4)x – 3]x + 2

by using the form of writing above, the polynomial value for x = 5 is
f(x) = [(2x + 4)x – 3]x + 2
f(5) = [(2 . 5 + 4)5 – 3]5 + 2
f(5) = [(10 + 4)5 – 3]5 + 2
f(5) = [14 . 5 + (– 3)]5 + 2
f(5) = [70 + (– 3)]5 + 2
f(5) = 67 . 5 + 2 = 335 + 2 = 337

The pattern seen in the above calculation is as follows,
Multiply (dot ".") 2 by 5, then add 4 then get 14.
Multiply 14 by 5, then add -3 then get 67.
Multiply 67 by 5, then add 2, then get 337.
The process can be presented with a more interesting scheme below,


Description: arrow indicates multiplied by 5


Example 2.
Calculate f (-2) for f(x) = 3x3 – 4x + 6.

Answer:
Using the schematic that has been studied above, arrange the numbers in the first row that contain each x coefficient from the highest rank down to the lower rank.

So, f(-2) = - 10

Description: because the coefficient of x2 does not exist, then where the coefficient in question is written zero.


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