Does any one know what ______ is?

Question

synthetic division


And also how to solve rational expressions IM me to help thank you

Answer 1

Answered this question in your other problem

Answer 2

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Synthetic Division: The Process (page 1 of 4)

Sections: Introduction, Worked examples, Finding zeroes, Factoring polynomials


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

If you are given, say, y = x2 + 5x + 6, you can factor this 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 x = –2 and x = –3 are the two zeroes of y.

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 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 you don't know, when working backwards from the zeroes, which quadratic you're working toward. For any non-zero value of "a", your quadratic will still have the same zeroes. This is a technical point; 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 use synthetic division.

Let's look again at the quadratic from above: y = x2 + 5x + 6 . From the Rational Roots Test, you know that ± 1, 2, 3, and 6 are possible zeroes of the quadratic. (And, from the factoring above, you know that the zeroes are –3 and –2.) How would you use synthetic division to check the potential zeroes? Well, think about how long divison works. If we 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 x2 + 5x + 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? Copyright © Elizabeth Stapel 2000-2006 All Rights Reserved

First, write the coefficients ONLY inside an upside-down division symbol; make sure you leave room underneath to write another row of numbers later:




Put the test zero, x = 1, at the left:




Carry down the first number, representing the leading coefficient:




Multiply this by the test zero, and carry the result up into the next column:




Add down the column:




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




Add down the column:





Note that we got the same result as when we did the long division:



The result is formatted differently, but you should recognize that, in each format, we arrived at the result "x + 6, with remainder 12".

You already know (from the factoring above) that x + 3 is a factor, and therefore that
x = –3 is a zero. 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, in a sense, 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.

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Answer 3

it is a method of dividing long algebric expressions expressed in polynomial form by another expression in polynomial form. You got to refer to math books for details because it is hard to type in here.

Answer 4

never heard of that and i am not the greatest person in math!

Answer 5

You're the guy that was in an exam, aren't you?

Answer 6

if it is a SAT question, fill in the blank

Answer 7

never heard of it but then again i am not that bright

Answer 8

it's an underscore

Answer 9

If you dont know, i cant tell you.