If x 1 x + 1 is a polynomial
The polynomial division causes great problems for many students. The following sections should help you to understand the polynomial division in a simple way. This article belongs to the field of math.
Polynomial division is a mathematical method used to calculate the zeros of polynomials. The method of calculation is similar to the written division that you already got to know in elementary school. For this reason, in the following we will first briefly discuss the written division and then apply this knowledge to the polynomial division.
Polynomial Division Video:
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Example: Written division (explanation below)
How did that work again? Here's how:
- The aim is to find the solution to exercise 840: 4
- The first number is 8. If you divide 8: 4 you get a 2. This is the first number for the answer
- Now count back again: 2 · 4 = 8. This 8 is written under the first 8 at the beginning.
- Now the two numbers are subtracted from each other, hence the "-" in front of the lower number. 8 - 8 equals 0.
- Now the next number is taken down from the top: This is a 4.
- Now it is shared again. 4: 4 = 1. The 1 is written down again
- Recalculation: 1 · 4 = 4. The 4 is written under the other 4 again
- Now it is deducted again: 4 - 4 = 0.
- The zero is written down. The next number is also pulled down from above, also a 0.
- 0: 4 = 0. A zero is appended to the result.
- Recalculation: 0 · 0 = 0 and 0 - 0 remains zero. There is no further figure to be obtained from above
- There are only zeros left. With that you have finished billing.
Polynomial division explanation
With the polynomial division, one does not divide two numbers, but whole terms. In mathematics, the term term describes a meaningful expression that can contain numbers, variables, symbols (for mathematical operations) and brackets. To perform a polynomial division, one needs a term and a zero of this term. To be precise, these are polynomials (what is a polynomial?). We divide one polynomial by another polynomial. That is why one speaks of a polynomial division.
Finding this zero is often quite difficult. In school, the teacher usually specifies the zero point. If this is not the case, a zero can be found by guessing or numerical methods. For the following examples, we assume that a zero is already given.Show:
Polynomial division example 1:
The function y = f (x) = x is given3 - 2x2 - 5x + 6. By trying, a zero was found at x = 1. The polynomial division should now be carried out in order to subsequently find all zeros.
Solution: We divide the function y = f (x) by (x - 1). It looks like this:
We first calculate x here3 : x = x2. Then we multiply x2 · (X - 1) = x3 - x2. Then (x3 - 2x2 ) - (x3 - x2 ) calculated. Then the game starts all over again until the division is complete. The procedure corresponds to the written division. The result of the polynomial division is x2 - x - 6. We can find out whether the result is correct through a test:
sample: (x2 - x - 6) (x - 1) = x3 - 2x2 -5x + 6 // The solution is right
In order to find the remaining zeros, we apply the PQ formula to x2 - x - 6 = 0 and get x2 = 3 and x3 = -2. So we know that the x-values 1, 3 and -2 have the zeros. The polynomial can thus be broken down into its linear factors: f (x) = (x - 1) (x - 3) (x + 2). Here, too, we conduct the trial:
sample: (x - 1) (x - 3) (x + 2) = x3 - 2x2 - 5x + 6 // The solution is right
Note: If you have problems recalculating here, you can read this in the article multiplying several brackets.
Polynomial division example 2
The function y = f (x) = 3x is given3 - 10x2 + 7x - 12. A zero point at x = 3 is known. Find all zeros of f (x).
Solution: We first divide the function f (x) by (x - 3). It looks like this:
Here, too, we calculate the result piece by piece. First is 3x3 : x calculated, the result is 3x2. We multiply back: 3x2 · (X - 3) and get 3x3 - 9x2. Then we subtract again and continue to calculate the task in the usual way. The result of the polynomial division is 3x2 - x + 4. We do a test to be on the safe side.
sample: (x - 3) (3x2 -x + 4) = 3x3 - 10x2 + 7x - 12
To find more zeros, we turn to 3x2 - x + 4 = 0 the PQ formula (important: first divide by 3). When using the PQ formula, you get a negative number under the root. This ends the calculation (for students) and the only zero is at x = 3.
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