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Synthetic division is a shorthand form of polynomial long division. Here is a brief description. Numerous videos and written descriptions are available from various web sites.
The tableau starts with the dividend (numerator) coefficients across the top in order by decreasing power from left to right. You will notice that your numerator is missing an x² term, so the coefficient of that is zero.
Either to the left or sligthly below those coefficients (and above a horizontal line), the root of the binomial divisor is shown. Here, your divisor is (x+3), so its root is x=-3. In the situation where the divisor is (ax+b), the root you will use is -b/a.
The leftmost number below the line is the leftmost dividend coefficient. It is simply brought straight down. Each successive number to the right of that above the line is found by multiplying the previous number below the line by the root. Here, -6 = (2)(-3), for example. The next number below the line is the sum of the polynomial coefficient value and this product. Here, (0) +(-6) = -6, for example.
The process of multipllying the sum by the root and adding to the next dividend coefficient proceeds left to right until you run out of dividend numbers. The final (rightmost) sum below the line is the remainder of the division, and also the value of the dividend polynomial at the "root". (The remainder will be zero if the divisor "root" is an actual root.) Here, the remainder is -40.
The result is a polynomial of degree one less than the original. Of course, any non-zero remainder is expressed as a fraction with the divisor as the denominator, just as it is with numerical long division. Here, the quotient is ...
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