Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor — and it *only* works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.

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If you are given, say, the polynomial equation *y* = *x*^{2} + 5x + 6, you can factor the polynomial as *y* = (*x* + 3)(*x* + 2). Then you can find the zeroes of *y* by setting each factor equal to zero and solving. You will find that the two zeroes of the polynomial are *x* = −2 and *x* = −3.

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You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that *x* = −2 and *x* = −3 are the zeroes of a quadratic, then you know that *x* + 2 = 0, so *x* + 2 is a factor, and *x* + 3 = 0, so *x* + 3 is a factor. Therefore, you know that the quadratic must be of the form *y* = *a*(*x* + 3)(*x* + 2).

(The extra number "*a*" in that last sentence is in there because, when you are working backwards from the zeroes, you don't know toward which quadratic you're working. For any non-zero value of "*a*", your quadratic will still have the same zeroes. But the issue of the value of "*a*" is just a technical consideration; as long as you see the relationship between the zeroes and the factors, that's all you really need to know for this lesson.)

Anyway, the above is a long-winded way of saying that, if *x* − *n* is a factor, then *x* = *n* is a zero, and if *x* = *n* is a zero, then *x* − *n* is a factor. And this is the fact you use when you do synthetic division.

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Let's look again at the quadratic from above: *y* = *x*^{2} + 5*x* + 6. From the Rational Roots Test, we know that ± 1, 2, 3, and 6 are possible zeroes of the quadratic. (And, from the factoring above, we know that the zeroes are, in fact, −3 and −2.) How would you use synthetic division to check the potential zeroes?

Well, think about how long polynomial divison works. If I were to guess that *x* = 1 is a zero, then this means that *x* − 1 is a factor of the quadratic. And if it's a factor, then it will divide out evenly; that is, if we divide *x*^{2} + 5*x* + 6 by *x* − 1, we would get a zero remainder. Let's check:

As expected (since we know that *x* − 1 is not a factor), we got a non-zero remainder. What does this look like in synthetic division?

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First, take the polynomial (in our case, *x*^{2} + 5*x* + 6), and write the coefficients ONLY inside an upside-down division-type symbol:

Make sure you leave room inside, underneath the row of coefficients, to write another row of numbers later.

Put the test zero, in our case *x* = 1, at the left, next to the (top) row of numbers:

Take the first number that's on the inside, the number that represents the polynomial's leading coefficient, and carry it down, unchanged, to below the division symbol:

Multiply this carry-down value by the test zero on the left, and carry the result up into the next column inside:

Add down the column:

Multiply the previous carry-down value by the test zero, and carry the new result up into the last column:

Add down the column:

This last carry-down value is the remainder.

Comparing, you can see that we got the same result from the synthetic division, the same quotient (namely, 1*x* + 6) and the same remainder at the end (namely, 12), as when we did the long division:

The results are formatted differently, but you should recognize that each format provided us with the same result, being a quotient of *x* + 6, and a remainder of 12.

We already know (from the factoring above) that *x* + 3 is a factor of the polynomial, and therefore that *x* = −3 is a zero.

Now let's compare the results of long division and synthetic division when we use the factor *x* + 3 (for the long division) and the zero *x* = −3 (for the synthetic division):

As you can see above, while the results are formatted differently, the results are otherwise the same:

In the long division, I divided by the factor *x* + 3, and arrived at the result of *x* + 2 with a remainder of zero. This means that *x* + 3 is a factor, and that *x* + 2 is left after factoring out the *x* + 3. Setting the factors equal to zero, I get that *x* = −3 and *x* = −2 are the zeroes of the quadratic.

In the synthetic division, I divided by *x* = −3, and arrived at the same result of *x* + 2 with a remainder of zero. Because the remainder is zero, this means that *x* + 3 is a factor and *x* = −3 is a zero. Also, because of the zero remainder, *x* + 2 is the remaining factor after division. Setting this equal to zero, I get that *x* = −2 is the other zero of the quadratic.

I will return to this relationship between factors and zeroes throughout what follows; the two topics are inextricably intertwined.

URL: https://www.purplemath.com/modules/synthdiv.htm

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