Octagon

In algebra, synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. It is mostly taught for division by binomials of the form

$x-a,$

but the method generalizes to division by any monic polynomial, and to any polynomial.

The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division. Also, the subtractions in long division are converted to additions by switching the signs at the very beginning, preventing sign errors.

Synthetic division for linear denominators is also called division through Ruffini's rule.

Regular synthetic division

The first example is synthetic division with only a monic linear denominator $x-a$ .

$\frac{x^3-12x^2-42}{x-3}$

Write the coefficients of the polynomial to be divided at the top (the zero is for the unseen 0x).

$\begin{array}{cc}\begin{array}{r}\\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& -12& 0& -42\\ & & & \end{array}}\end{array}$

Negate the coefficients of the divisor.

$\begin{array}{rr}-1x& +3\end{array}$

Write in every coefficient of the divisor but the first one on the left.

$\begin{array}{cc}\begin{array}{r}\\ 3\\ \end{array}& \overline{)\begin{array}{rrrr}1& -12& 0& -42\\ & & & \end{array}}\end{array}$

Note the change of sign from −3 to 3. "Drop" the first coefficient after the bar to the last row.

$\begin{array}{cc}\begin{array}{r}\\ 3\\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& -12& 0& -42\\ & & & \\ 1& & & \\ \end{array}}\end{array}$

Multiply the dropped number by the number before the bar, and place it in the next column.

$\begin{array}{cc}\begin{array}{r}\\ 3\\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& -12& 0& -42\\ & 3& & \\ 1& & & \\ \end{array}}\end{array}$

Perform an addition in the next column.

$\begin{array}{cc}\begin{array}{c}\\ 3\\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& -12& 0& -42\\ & 3& & \\ 1& -9& & \\ \end{array}}\end{array}$

Repeat the previous two steps and the following is obtained:

$\begin{array}{cc}\begin{array}{c}\\ 3\\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& -12& 0& -42\\ & 3& -27& -81\\ 1& -9& -27& -123\end{array}}\end{array}$

Count the terms to the left of the bar. Since there is only one, the remainder has degree zero. Mark the separation with a vertical bar.

$\overline{)\begin{array}{rrrr}1& -9& -27& -123\end{array}}$

The terms are written with increasing degree from right to left beginning with degree zero for both the remainder and the result.

$\overline{)\begin{array}{rrrr}1x^2& -9x& -27& -123\end{array}}$

The result of our division is:

$\frac{x^3-12x^2-42}{x-3}=x^2-9x-27-\frac{123}{x-3}$

Evaluating Polynomials by the Remainder Theorem

The above form of synthetic division is useful in the context of the Polynomial remainder theorem for evaluating univariate polynomials. To summarize, the value of $p(x)$ at $a$ is equal to the remainder of $\frac{p(x)}{(x-a)}$. The advantage of calculating the value this way is that it requires just over half as many multiplication steps as naive evaluation. An alternative evaluation strategy is Horner's method.

Expanded synthetic division

This method generalizes to division by any monic polynomial with only a slight modification with changes in bold. Using the same steps as before, let's try to perform the following division:

$\frac{x^3-12x^2-42}{x^2+x-3}$

We concern ourselves only with the coefficients. Write the coefficients of the polynomial to be divided at the top.

$\overline{)\begin{array}{rrrr}1& \text{-}12& 0& \text{-}42\end{array}}$

Negate the coefficients of the divisor.

$\begin{array}{rrr}\text{-}1x^2& -1x& +3\end{array}$

Write in every coefficient but the first one on the left in an upward right diagonal (see next diagram).

$\begin{array}{cc}\begin{array}{r}\\ & 3\\ \text{-}1& \\ \end{array}& \overline{)\begin{array}{rrrr}1& \text{-}12& 0& \text{-}42\\ & & & \\ & & & \end{array}}\end{array}$

Note the change of sign from 1 to −1 and from −3 to 3 . "Drop" the first coefficient after the bar to the last row.

$\begin{array}{cc}\begin{array}{r}\\ & 3\\ \text{-}1& \\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& \text{-}12& 0& \text{-}42\\ & & & \\ & & & \\ 1& & & \\ \end{array}}\end{array}$

Multiply the dropped number by the diagonal before the bar, and place the resulting entries diagonally to the right from the dropped entry.

$\begin{array}{cc}\begin{array}{r}\\ & 3\\ \text{-}1& \\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& \text{-}12& 0& \text{-}42\\ & & 3& \\ & \text{-}1& & \\ 1& & & \\ \end{array}}\end{array}$

Perform an addition in the next column.

$\begin{array}{cc}\begin{array}{r}\\ & 3\\ \text{-}1& \\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& \text{-}12& 0& \text{-}42\\ & & 3& \\ & \text{-}1& & \\ 1& \text{-}13& & \\ \end{array}}\end{array}$

Repeat the previous two steps until you would go past the entries at the top with the next diagonal.

$\begin{array}{cc}\begin{array}{r}\\ & 3\\ \text{-}1& \\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& \text{-}12& 0& \text{-}42\\ & & 3& \text{-}39\\ & \text{-}1& 13& \\ 1& \text{-}13& 16& \\ \end{array}}\end{array}$

Then simply add up any remaining columns.

$\begin{array}{cc}\begin{array}{r}\\ & 3\\ \text{-}1& \\ \\ \end{array}& \overline{)\begin{array}{rrrr}1& \text{-}12& 0& \text{-}42\\ & & 3& \text{-}39\\ & \text{-}1& 13& \\ 1& \text{-}13& 16& \text{-}81\\ \end{array}}\end{array}$

Count the terms to the left of the bar. Since there are two, the remainder has degree one. Mark the separation with a vertical bar.

$\overline{)\begin{array}{rrrr}1& \text{-}13& 16& \text{-}81\end{array}}$

The terms are written with increasing degree from right to left beginning with degree zero for both the remainder and the result.

$\overline{)\begin{array}{rrrr}1x& \text{-}13& 16x& \text{-}81\end{array}}$

The result of our division is:

$\frac{x^3-12x^2-42}{x^2+x-3}=x-13+\frac{16x-81}{x^2+x-3}$

For non-monic divisors

With a little prodding, the expanded technique may be generalised even further to work for any polynomial, not just monics. The usual way of doing this would be to divide the divisor $g(x)$ with its leading coefficient (call it a):

$h(x)=\frac{g(x)}{a}$

then using synthetic division with $h(x)$ as the divisor, and then dividing the quotient by a to get the quotient of the original division (the remainder stays the same). But this often produces unsightly fractions which get removed later, and is thus more prone to error. It is possible to do it without first dividing the coefficients of $g(x)$ by a.

As can be observed by first performing long division with such a non-monic divisor, the coefficients of $f(x)$ are divided by the leading coefficient of $g(x)$ after "dropping", and before multiplying.

Let's illustrate by performing the following division:

$\frac{6x^3+5x^2-7}{3x^2-2x-1}$

A slightly modified table is used:

$\begin{array}{c}\begin{array}{r}\\ & 1& \\ 2& & \\ \\ & & /3\\ \end{array}\overline{)\begin{array}{rrrr}6& 5& 0& \text{-}7\\ & & & \\ & & & \\ & & & \\ & & & \\ \end{array}}\end{array}$

Note the extra row at the bottom. This is used to write values found by dividing the "dropped" values by the leading coefficient of $g(x)$ (in this case, indicated by the /3; note that, unlike the rest of the coefficients of $g(x)$, the sign of this number is not changed).

Next, the first coefficient of $f(x)$ is dropped as usual:

$\begin{array}{c}\begin{array}{r}\\ & 1& \\ 2& & \\ \\ & & /3\\ \end{array}\overline{)\begin{array}{rrrr}6& 5& 0& \text{-}7\\ & & & \\ & & & \\ 6& & & \\ & & & \\ \end{array}}\end{array}$

and then the dropped value is divided by 3 and placed in the row below:

$\begin{array}{c}\begin{array}{r}\\ & 1& \\ 2& & \\ \\ & & /3\\ \end{array}\overline{)\begin{array}{rrrr}6& 5& 0& \text{-}7\\ & & & \\ & & & \\ 6& & & \\ 2& & & \\ \end{array}}\end{array}$

Next, the new (divided) value is used to fill the top rows with multiples of 2 and 1, as in the expanded technique:

$\begin{array}{c}\begin{array}{r}\\ & 1& \\ 2& & \\ \\ & & /3\\ \end{array}\overline{)\begin{array}{rrrr}6& 5& 0& \text{-}7\\ & & 2& \\ & 4& & \\ 6& & & \\ 2& & & \\ \end{array}}\end{array}$

The 5 is dropped next, with the obligatory adding of the 4 below it, and the answer is divided again:

$\begin{array}{c}\begin{array}{r}\\ & 1& \\ 2& & \\ \\ & & /3\\ \end{array}\overline{)\begin{array}{rrrr}6& 5& 0& \text{-}7\\ & & 2& \\ & 4& & \\ 6& 9& & \\ 2& 3& & \\ \end{array}}\end{array}$

Then the 3 is used to fill the top rows:

$\begin{array}{c}\begin{array}{r}\\ & 1& \\ 2& & \\ \\ & & /3\\ \end{array}\overline{)\begin{array}{rrrr}6& 5& 0& \text{-}7\\ & & 2& 3\\ & 4& 6& \\ 6& 9& & \\ 2& 3& & \\ \end{array}}\end{array}$

At this point, if, after getting the third sum, we were to try and use it to fill the top rows, we would "fall off" the right side, thus the third sum is the first coefficient of the remainder, as in regular synthetic division. But the values of the remainder are not divided by the leading coefficient of the divisor:

$\begin{array}{c}\begin{array}{r}\\ & 1& \\ 2& & \\ \\ & & /3\\ \end{array}\overline{)\begin{array}{rrrr}6& 5& 0& \text{-}7\\ & & 2& 3\\ & 4& 6& \\ 6& 9& 8& \text{-}4\\ 2& 3& & \\ \end{array}}\end{array}$

Now we can read off the coefficients of the answer. As in expanded synthetic division, the last two values (2 is the degree of the divisor) are the coefficients of the remainder, and the remaining values are the coefficients of the quotient:

$\overline{)\begin{array}{rrrr}2x& +3& 8x& \text{-}4\end{array}}$

and the result is

$\frac{6x^3+5x^2-7}{3x^2-2x-1}=2x+3+\frac{8x-4}{3x^2-2x-1}$

Compact Expanded Synthetic Division

However, the diagonal format above becomes less space-efficient when the degree of the divisor exceeds half of the degree of the dividend. It is easy to see that we have complete freedom to write each product in any row, as long as it is in the correct column. So the algorithm can be compactified by a greedy strategy, as illustrated in the division below.

$\frac{a{x}^7+b{x}^6+c{x}^5+d{x}^4+e{x}^3+f{x}^2+gx+h}{i{x}^4-j{x}^3-k{x}^2-lx-m}=n{x}^3+o{x}^2+px+q+\frac{r{x}^3+s{x}^2+tx+u}{i{x}^4-j{x}^3-k{x}^2-lx-m}$

$\begin{array}{cc}\begin{array}{r}\\ \\ \\ \\ j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}& & & & qj& & & \\ & & & pj& pk& qk& & \\ & & oj& ok& ol& pl& ql& \\ & nj& nk& nl& nm& om& pm& qm\\ a& b& c& d& e& f& g& h\\ a& o_0& p_0& q_0& r& s& t& u\\ n& o& p& q& & & & \\ \end{array}}\end{array}$

The following describes how to perform the algorithm; this algorithm includes steps for dividing non-monic divisors:

1. Write the coefficients of the dividend on a bar
   
   $\begin{array}{c}\overline{)\begin{array}{rrrrrrrr}a& b& c& d& e& f& g& h\end{array}}\end{array}$
   
   
2. Negate the coefficients of the divisor. Write in every coefficient of the divisor but the first (leading coefficient) one on the left.
   
   $\begin{array}{cc}\begin{array}{rrrr}j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}a& b& c& d& e& f& g& h\end{array}}\end{array}$
   
   
3. From the number of coefficients placed on the left side, count the number of dividend coefficients above the bar, starting from the rightmost column. Then place a vertical bar on the row below and to the left of that column. This vertical bar marks the separation between the quotient and the remainder.
   
   $\begin{array}{cc}\begin{array}{rrrr}j& k& l& m\\ \\ \end{array}& \overline{)\begin{array}{rrrrrrrr}a& b& c& d& e& f& g& h\\ & & & & & & & \\ \end{array}}\end{array}$
   
   
4. Drop the first coefficient of the dividend below the bar.
   
   $\begin{array}{cc}\begin{array}{rrrr}j& k& l& m\\ \\ \end{array}& \overline{)\begin{array}{rrrrrrrr}a& b& c& d& e& f& g& h\\ a& & & & & & & \\ \end{array}}\end{array}$
   
   
5. • Divide the last dropped/summed number by the leading coefficient of the divisor and place it on the row below (this doesn't need to be done if the coefficient is 1). In this case $n=\frac{a}{i}$
   • Multiply the last dropped/summed number (or the divided dropped/summed number) to each negated coefficients on the left (starting with the left most); skip if the summed number is zero. Place each product on top of the subsequent columns.

   $\begin{array}{cc}\begin{array}{r}\\ j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}& nj& nk& nl& nm& & & \\ a& b& c& d& e& f& g& h\\ a& & & & & & & \\ n& & & & & & & \\ \end{array}}\end{array}$

6. Perform an column-wise addition on the next column.
   $\begin{array}{cc}\begin{array}{r}\\ j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}& nj& nk& nl& nm& & & \\ a& b& c& d& e& f& g& h\\ a& o_0& & & & & & \\ n& & & & & & & \\ \end{array}}\end{array}$
7. Repeat the previous two steps. Stop when you performed the previous two steps on the number just before the vertical bar.
   
   Let $o=\frac{o_0}{i}$
   $\begin{array}{cc}\begin{array}{r}\\ \\ j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}& & oj& ok& ol& & & \\ & nj& nk& nl& nm& om& & \\ a& b& c& d& e& f& g& h\\ a& o_0& p_0& & & & & \\ n& o& & & & & & \\ \end{array}}\end{array}$
   
   Let $p=\frac{p_0}{i}$
   $\begin{array}{cc}\begin{array}{r}\\ \\ \\ j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}& & & pj& pk& & & \\ & & oj& ok& ol& pl& & \\ & nj& nk& nl& nm& om& pm& \\ a& b& c& d& e& f& g& h\\ a& o_0& p_0& q_0& & & & \\ n& o& p& & & & & \\ \end{array}}\end{array}$
   
   Let $q=\frac{q_0}{i}$
   $\begin{array}{cc}\begin{array}{r}\\ \\ \\ \\ j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}& & & & qj& & & \\ & & & pj& pk& qk& & \\ & & oj& ok& ol& pl& ql& \\ & nj& nk& nl& nm& om& pm& qm\\ a& b& c& d& e& f& g& h\\ a& o_0& p_0& q_0& r& & & \\ n& o& p& q& & & & \\ \end{array}}\end{array}$
8. Perform the remaining column-wise additions on the subsequent columns (getting the remainder).
   $\begin{array}{cc}\begin{array}{r}\\ \\ \\ \\ j& k& l& m\\ \end{array}& \overline{)\begin{array}{rrrrrrrr}& & & & qj& & & \\ & & & pj& pk& qk& & \\ & & oj& ok& ol& pl& ql& \\ & nj& nk& nl& nm& om& pm& qm\\ a& b& c& d& e& f& g& h\\ a& o_0& p_0& q_0& r& s& t& u\\ n& o& p& q& & & & \\ \end{array}}\end{array}$
9. The results below the horizontal bar would be interpreted with increasing degree from right to left beginning with degree zero for both the remainder and the result.
   
   $\frac{a{x}^7+b{x}^6+c{x}^5+d{x}^4+e{x}^3+f{x}^2+gx+h}{i{x}^4-j{x}^3-k{x}^2-lx-m}=n{x}^3+o{x}^2+px+q+\frac{r{x}^3+s{x}^2+tx+u}{i{x}^4-j{x}^3-k{x}^2-lx-m}$
   
   

See also

• Polynomial remainder theorem
• Euclidean domain
• Gröbner basis
• Greatest common divisor of two polynomials
• Horner scheme

References

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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang