can someone please explain synthetic division to me?

It's just a specialized way to divide a polynomial by a linear factor using long division. I don't see the reason why anyone would want to learn this technique, because it's just long division.

If you type "synthetic division" in google you'll find websites explaining the technique in detail.

It is like long division, but with algebraic polynomials instead of numbers. I'd need a graphics whiteboard to illustrate it. I suggest you read one of the online explanations such as
http://www.purplemath.com/modules/synthdiv.htm

So is it any different than long division? I just wanted someone to explain it because we are studying it in algebra and the exa,moples weren't making sense to me. Thank you.

Yes, synthetic division is a specialized and more efficient method for dividing polynomials by linear factors compared to long division.

Here's a step-by-step explanation of synthetic division:

1. First, you need to set up the problem correctly. Let's say you have a polynomial expression of the form:

P(x) = ax^n + bx^(n-1) + cx^(n-2) + ... + kx + l

And you want to divide it by a linear factor of the form:

(x - r)

The objective is to find the quotient polynomial Q(x) and the remainder term R (if any).

2. To set up the synthetic division table, write down the coefficients of the polynomial expression P(x) in decreasing order of powers of x. If there are missing terms, include zeros as placeholders.

For example, if P(x) = 3x^3 + 2x^2 - 5x + 1, the table would look like:

| 3 2 -5 1
r |___________________
|

3. Next, write down the constant term, which is the negation of the value inside the linear factor (x - r). So, if r is 2, write -2 in the bottom row of the table.

| 3 2 -5 1
r |_____________ -2
|

4. Now, bring down the first coefficient (3 in this example) to the bottom row of the table.

| 3 2 -5 1
r |_____________ -2
| 3

5. Multiply r (in this case, -2) by the number at the bottom (3) and write the result under the next coefficient (2) in the top row. Then, add the two numbers and write the sum below.

| 3 2 -5 1
r |_____________ -2
| 3 -6

6. Repeat the previous step for the remaining coefficients in the top row, always multiplying r by the number at the bottom and adding the result to the next coefficient.

| 3 2 -5 1
r |_____________ -2
| 3 -6 2

7. Finally, the numbers in the bottom row represent the coefficients of the quotient polynomial Q(x), except the last number (-2 in this example) is the remainder (R).

So, Q(x) = 3x^2 - 6x + 2 and R = -2.

That's the basic idea of synthetic division. It simplifies the process of dividing polynomials by linear factors by avoiding some of the repetitive calculations required in long division. Practice with some examples to become more comfortable with the technique.