Synthetic division, also known as Ruffini's rule, is a shortcut method for dividing a polynomial (a mathematical expression with two or more terms) by a binomial (a mathematical expression with two terms). It's called "synthetic" because it's a quick way to do division without actually writing out the long division problem.
Let's say we have a polynomial expression, like "3x^3 + 5x^2 - 2x + 7", and we want to divide it by "x - 2". To do this using synthetic division, we first write down the polynomial and the binomial in a special way:
3 5 -2 7 | x^3 + x^2 - 2x + 7 |********_******** | x - 2 |********_********
Next, we divide the first term of the polynomial (3x^3) by the first term of the binomial (x). This gives us 3. We then write the 3 in the top row, under the division sign.
Next, we multiply the binomial (x - 2) by the 3 that we just found. This gives us "3x - 6". We write this product under the polynomial expression.
Next, we add the polynomial and the product, and write the result in the next line.
3 5 -2 7 | x^3 + x^2 - 2x + 7 |********_******** | x - 2 |********_******** 3 2x^2 - 2x + 1
We repeat this process for the next term in the polynomial (5x^2), dividing it by x and then multiplying it by the binomial. We then add this to the result, and repeat the process for each term of the polynomial.
3 5 -2 7 | x^3 + x^2 - 2x + 7 |********_******** | x - 2 |********_******** 3 2x^2 - 2x + 1 |_ | -10x + 13 |__** | -3x + 4 |**_** | 11 |**_****
After we've done the last step, we can see that the answer to our division problem is "3x^2 - 2x + 1, with a remainder of 11".
So, synthetic division is a quick and easy way to divide a polynomial by a binomial. Instead of writing out the long division problem, we just write the polynomial and binomial in a special way, divide the first term of the polynomial by the first term of the binomial, multiply the binomial by that answer, add the result to the next term of the polynomial, and repeat the process until we've divided all the terms.