If you already understand Polynomial Long Division perfectly, you might want to try Synthetic Long Division. It is a different method through which you can divide polynomials. However, it is much shorter than Polynomial Long Division. I’m not sure if you have seen it or will see it with Mr. Alcantara, but it's still a good method to know.
Differences between Synthetic and Polynomial Long Division
Synthetic Division only applies to polynomials that can be divided into polynomials of the form x-a, or monic 1st-degree polynomials.
For instance,
(x^2-2x-3) ÷ (x+1) can be done by synthetic division because the divisor is x+1, a polynomial in the form x-a, in which a = -1.
(x^3-3x^2+x-3) ÷ (x^2+1) cannot be solved through synthetic division because the divisor is x^2+1, which is not in the form of x-a.
Differences between Synthetic and Polynomial Long Division
Synthetic Division only applies to polynomials that can be divided into polynomials of the form x-a, or monic 1st-degree polynomials.
For instance,
(x^2-2x-3) ÷ (x+1) can be done by synthetic division because the divisor is x+1, a polynomial in the form x-a, in which a = -1.
(x^3-3x^2+x-3) ÷ (x^2+1) cannot be solved through synthetic division because the divisor is x^2+1, which is not in the form of x-a.
Synthetic Division only shows the coefficients of terms, while Polynomial Long Division shows the entire polynomial.
Synthetic Division
Setup
*If you have been taught this by an American teacher, you might setup your problem different to the way I explain here. I learned it the Colombian way with Mr. Troncoso.
As with Polynomial Long Division, I will use a problem to show you the process.
(x^2-2x-3) ÷ (x+1)
Our setup will only show the coefficients of the terms. This means that instead of showing x^2, it will show 1; and instead of showing -2x, it will show -2.
In this sense, the first thing we must do to setup our problem, is write down the coefficients of the terms of the dividend, in order of decreasing degree.
1 -2 -3
*Just like in Polynomial Long Division, we must also include coefficients of all terms. For instance, if we were to write the coefficients of x^3+4x, we would write 1 0 4 0. In our problem, however, there are no coefficients of zero.
The general setup for synthetic division looks like this:
Synthetic Division
Setup
*If you have been taught this by an American teacher, you might setup your problem different to the way I explain here. I learned it the Colombian way with Mr. Troncoso.
As with Polynomial Long Division, I will use a problem to show you the process.
(x^2-2x-3) ÷ (x+1)
Our setup will only show the coefficients of the terms. This means that instead of showing x^2, it will show 1; and instead of showing -2x, it will show -2.
In this sense, the first thing we must do to setup our problem, is write down the coefficients of the terms of the dividend, in order of decreasing degree.
1 -2 -3
*Just like in Polynomial Long Division, we must also include coefficients of all terms. For instance, if we were to write the coefficients of x^3+4x, we would write 1 0 4 0. In our problem, however, there are no coefficients of zero.
The general setup for synthetic division looks like this:
where a is found in the divisor, which is in the form of x-aTherefore, the setup for our problem should look like this.
After we’ve setup our problem, we’re ready to begin.1. The first thing we must do is bring down the first coefficient in the list. By bringing down, we mean copying it below the horizontal line, like shown below.
2. Then, we multiply the number we just brought down by a, or, in this case, -1.1*(-1) = -1
3. We will now write the product below the next coefficient in line, but above the horizontal line.
4. Now, we will add the product we just wrote down and the number directly above it.-2+(-1) = -3
5. We now write the sum below the numbers we added, below the horizontal line.
6. Repeat steps 2-5 until we run out of numbers.
-3+3 = 0
7. Our answer is the one written below the horizontal line. However, like we saw before, the method only shows coefficients. This means that the numbers below the horizontal line are the coefficients of our answer. To write our answer, we will start from the right.The first number on the right is the remainder, which in this case is 0. The remainder is always written as a fraction, with the right-most number as the numerator and the divisor as your denominator. In this case, it will look like this:
0 ÷ (x+1) = 0, which makes sense since we already said there was no remainder for this problem.
The number directly next to the left (in this case, -3) is the coefficient of x^0; the number that follows to the left (in this case, 1) is the coefficient of x^1; and it goes on like that always increasing in degree as you move towards the left.
We can now write our answer:
1*x^1-3*X^0+0 =
x-3
This means that
(x^2-2x-3) ÷ (x+1) = x-3
Note: In the expression (x-a) explained above, a may also be thought as the solution to the function, or the zeros in the graph of the function.
For example,
(x^2-2x-3) ÷ (x+1) = x-3, where a = -1, can also be written like this:
x^2-2x-3 =
(x+1)(x-3)
Looking for the solution to this problem is the same as looking for the values of x that make it zero.
(x+1)(x-3) = 0
If we want (x+1)(x-3) to equal zero, either x+1 or x-3 must equal zero because anything multiplied by zero equals zero. Therefore, the solutions are:
x+1 = 0
x = -1
and
x-3 = 0
x = 3
In our problem, a = -1, which is one of our solutions.
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