Use Synthetic Division to Solve What is the Quotient

Constructed Division: The Process

How are polynomial zeroes and factors related?

If you are given, say, the polynomial equation

y
=10
² + 5x + vi, you can factor the polynomial equally

y
= (x + iii)(x + 2). Then you can find the zeroes of

y

by setting each gene equal to zero and solving. You will find that the two zeroes of the polynomial are

10 = −two
and

ten = −3.

You can, yet, besides work backwards from the zeroes to discover the originating polynomial. For instance, if y’all are given that

x = −2
and

x = −iii
are the zeroes of a quadratic, and so you know that

x + 2 = 0, so

x + 2
is a factor, and

x + 3 = 0, and then

x + iii
is a factor. Therefore, you know that the quadratic must exist of the form

y
=
a(x + 3)(ten + ii).

(The actress number “
a
” in that last sentence is in there considering, when you are working backwards from the zeroes, you don’t know toward which quadratic y’all’re working. For any non-zero value of “
a
“, your quadratic volition notwithstanding have the same zeroes. But the issue of the value of “
a
” is merely a technical consideration; as long as you lot see the relationship between the zeroes and the factors, that’s all you lot really need to know for this lesson.)

Anyhow, the above is a long-winded style of saying that, if

x −north

is a factor, and so

10 =due north

is a nada, and if

x =north

is a nil, and then

x −north

is a factor. And this is the fact you use when you lot do synthetic division.

Let’s look again at the quadratic from above:

y
=ten
² + 5x + 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 y’all use synthetic division to check the potential zeroes?

Well, think near how long polynomial divison works. If I were to guess that

x = 1
is a nix, so this means that

x − one
is a cistron of the quadratic. And if it’s a cistron, and then information technology will carve up out evenly; that is, if we dissever

x
^two + 5x + 6
past

10 − one, we would go a cypher residuum. Let’southward cheque:

Equally expected (since nosotros know that

x − 1
is non a factor), nosotros got a non-nada residual. What does this look like in synthetic segmentation?

---

How do you practice synthetic sectionalization?

First, take the polynomial (in our example,

ten
² + 5x + 6), and write the coefficients ONLY inside an upside-downwards segmentation-type symbol:

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

Put the examination zero, in our case

10 = 1, at the left, next to the (top) row of numbers:

Accept the commencement number that’south on the inside, the number that represents the polynomial’s leading coefficient, and carry it down, unchanged, to below the division symbol:

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

Add downward the column:

Multiply the previous acquit-downwards value by the test zero, and deport the new result up into the last column:

Add down the column:

This last comport-down value is the remainder.

Comparing, yous can see that we got the same issue from the synthetic partition, the same caliber (namely,
ix + 6) and the same residue at the stop (namely,
12), as when we did the long division:

The results are formatted differently, merely you should recognize that each format provided united states of america with the same outcome, being a quotient of

x + 6, and a rest of
12.

---

Nosotros already know (from the factoring to a higher place) that

10 + three
is a factor of the polynomial, and therefore that

10 = −3
is a zero.

Now allow’s compare the results of long division and synthetic sectionalisation when we employ the factor

x + 3
(for the long partition) and the nada

x = −three
(for the synthetic division):

Equally you can encounter above, while the results are formatted differently, the results are otherwise the same:

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