Synthetic division or Long division of polynomials?

In summary, synthetic division and regular polynomial division both give the same results, but are two different methods for dividing polynomials. Synthetic division only works for dividing by a linear factor, while regular division can be used for any polynomial. Synthetic division is a simplified form of division for linear factors, and there is a generalization for arbitrary polynomials. These two methods often work together, as seen in the example of finding the factors of (8x^6 + 7x^3 -1).
  • #1
streetmeat
8
0
How do i know under which circumstances to use synthetic and when to just do regular polynomial division? do they not both give the same results?
 
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  • #2
Yes, they give the same results. They are just 2 different methods for the same thing, I prefer the long division though.
 
  • #3
Synthetic division only works if you are dividing a polynomial by a linear factor..
 
  • #4
In fact, only when dividing by something of the form x-a.

Synthetic division is just a simplified way of writing a division of that very special (but very important) form.
 
  • #5
derekjn said:
Synthetic division only works if you are dividing a polynomial by a linear factor..

There is a generalization to arbitrary polynomials
 
  • #6
http://www.pims.math.ca/pi/issue7/page13-16.pdf [Broken]
 
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  • #7
Hi i find them to go together, hand in hand.

for instance, we have this problem: 8x^6 + 7x^3 -1

i use synthetic division to find that -1 is a solution, hence i have a factor that is:
(x-1)

now, to look for the rest of the factors, i use long division to divide (8x^6 + 7x^3 -1)
by (x-1);


:)
 

1. What is the difference between synthetic division and long division of polynomials?

Synthetic division is a method used to divide polynomials by a binomial of the form (x - a). It is a quicker and more efficient method compared to long division, which involves dividing each term of the polynomial by the divisor and then subtracting the remainder.

2. How do I know when to use synthetic division or long division for polynomial division?

Synthetic division is only applicable when dividing by a binomial of the form (x - a). If the divisor is not in this form, then long division should be used. Additionally, synthetic division can only be used when the degree of the polynomial being divided is one more than the degree of the divisor.

3. What are the steps for performing synthetic division?

The steps for synthetic division are as follows:
1. Set up the division problem by writing the coefficients of the polynomial in descending order.
2. Write the constant term of the divisor on the left side of the division bar.
3. Bring down the first coefficient of the dividend (the polynomial being divided) and write it on the right side of the division bar.
4. Multiply the constant term of the divisor by the coefficient just brought down and write the result below the next coefficient of the dividend.
5. Add this new term to the next coefficient of the dividend and write the result below.
6. Repeat this process until all coefficients of the dividend have been used. The final result will be the quotient.

4. Is synthetic division always accurate?

Yes, synthetic division always gives the same result as long division. However, it is important to note that synthetic division can only be used for specific types of polynomial division, as mentioned in the answer to question 2.

5. Can synthetic division be used for polynomial division with remainders?

No, synthetic division can only be used for division problems with no remainders. If there is a remainder, long division must be used to accurately solve the problem.

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