Synthetic Division and the Binomial Expansion: A Comprehensive Guide | Study notes Mathematics

SYNTHETICDIVISION

Generalstepsfordividingsynthetically:

1.Arrangethecoefficientsofthepolynomialtobedividedinorderofdescendingpowersof

thevariable,supplyingzeroasthecoefficientofeachmissingpower.

2.Usetheconstanttermofthedivisorwithitssignchanged.Note:Thecoefficientandthe

powerofthevariabletermofthedivisormustbe1.(Example:2or6)

3.Bringdownthecoefficientofthelargestpowerof,multiplyitbythedivisor,placethe

productbeneaththecoefficientofthesecondlargestterm,andadd.Multiplythesumby

thedivisor,andplacetheproductbeneaththenextlargestpowerof.Continuethis

procedureuntilthereisaproductaddedtotheconstantterm.

4.Thelastnumberinthethirdrowistheremainder,andtheothernumbers,readingfromleft

toright,arethecoefficientsofthequotient,whichisofonedegreelessthanthegiven

polynomial.

Example:

2  32 2 7 10

22710     4  6

24 2 3 4 

3  10

3  6

   4



Answer:2  3  





Followingtheschematic:Theconstanttermofthedivisor,withitssignchanged,isplaced

outsidethedividend.Thecoefficientsofthedividendareplaced

indescendingorderaccordingtothepowerofthevariable.Bring

downthe2,multiplyitbythedivisor(‐2),placethe‐4belowthe7,

andadd.Multiplythe3bythe‐2,placethe‐6belowthe10,and

add.Theremainderis4.

Usesyntheticdivisiontofindthequotientandanyremainders.

1.󰇛5

28

󰇜󰇛1

󰇜

2.󰇛

1

󰇜󰇛1

󰇜

3.󰇛32

72

󰇜󰇛2

󰇜

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SYNTHETIC DIVISION

General steps for dividing synthetically:

1. Arrange the coefficients of the polynomial to be divided in order of descending powers of

the variable, supplying zero as the coefficient of each missing power.

2. Use the constant term of the divisor with its sign changed. Note: The coefficient and the

power of the variable term of the divisor must be 1. (Example: ݔ൅ 2 or ݔെ 6)

3. Bring down the coefficient of the largest power of ݔ, multiply it by the divisor, place the

product beneath the coefficient of the second largest term, and add. Multiply the sum by

the divisor, and place the product beneath the next largest power of ݔ. Continue this

procedure until there is a product added to the constant term.

4. The last number in the third row is the remainder, and the other numbers, reading from left

to right, are the coefficients of the quotient, which is of one degree less than the given

polynomial.

Example:

ݔ൅ 2 ݔ2 ଶ^ ൅ 7 ݔ൅ 10 െ4 െ 6

ݔ2 ଶ^ ݔ4 ൅ 2 3 4

Answer: 2 ݔ൅ 3 ൅

ସ ௫ାଶ

Following the schematic: The constant term of the divisor, with its sign changed, is placed

outside the dividend. The coefficients of the dividend are placed

in descending order according to the power of the variable. Bring

down the 2, multiply it by the divisor(‐2), place the ‐ 4 below the 7,

and add. Multiply the 3 by the ‐2, place the ‐ 6 below the 10, and

add. The remainder is 4.

Use synthetic division to find the quotient and any remainders.

1. ݔሺ ଷ^ ݔ5 െ ଶ^ ൅ 2 ݔ൅ 8ሻ ൊ ሺ ݔെ 1ሻ

2. ݔሺ ସ^ ݔ െ ଷ^ ൅ ݔെ 1ሻ ൊ ሺ ݔെ 1ሻ

3. ݔሺ3 ସ^ ݔ2 െ ଶ^ െ 7 ݔെ 2ሻ ൊ ሺ ݔെ 2ሻ

THE BINOMIAL EXPANSION This handout shows how to write and apply the formula for the expansion of expressions of the form ሺ ݔ൅ ݕሻ ௡^ where ݊ is any positive integer. In order to write the formula, we must generalize the information in the following chart: ሺ ݔ൅ ݕሻଵ^ ൌ ݕ ൅ ݔ ሺ ݔ൅ ݕሻ ଶ^ ൌ ݔ ଶ^ ݕ ൅ ݕݔ2 ൅ ଶ ሺ ݔ൅ ݕሻ ଷ^ ൌ ݔ ଷ^ ݔ3 ൅ ଶ^ ݕݔ3 ൅ ݕ ଶ^ ݕ ൅ ଷ ሺ ݔ൅ ݕሻ ସ^ ൌ ݔ ସ^ ݔ4 ൅ ଷ^ ݔ6 ൅ ݕ ଶ^ ݕ ଶ^ ݕݔ4 ൅ ଷ^ ݕ ൅ ସ ሺ ݔ൅ ݕሻ ହ^ ݔ ൌ ହ^ ݔ5 ൅ ସ^ ݕ൅ 10ݔ ଷ^ ݕ ଶ^ ൅ 10ݔ ଶ^ ݕ ଷ^ ݕݔ5 ൅ ସ^ ݕ ൅ ହ Note: The polynomials to the right have been found by expanding the binomials on the left, we just haven’t shown the work. There are a number of similarities to notice among the polynomials on the right. Here is a list of them:

In each polynomial, the sequence of exponents on the variable ݔ decreases to zero from the exponent on the binomial on the left. (The exponent 0 is not shown, since ݔ ଴^ ൌ 1).
In each polynomial, the exponents on the variable ݕ increase from 0 to the exponent on the binomial on the left. (Since ݕ ଴^ ൌ 1, it is not shown in the first term).
The sum of the exponents on the variables in any single term is equal to the exponent on the binomial to the left. The pattern in the coefficients of the polynomials in the right can best be seen by writing the right side again without the variables. It looks like this: 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 This triangular‐shaped array of coefficients is called Pascal’s Triangle. Each entry in the triangular array is obtained by adding the two numbers above it. Each row begins and ends with the same number, 1. If we were to continue Pascal’s Triangle, the next two rows would be: 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 Pascal’s Triangle can be used to find coefficients for the expansion ofሺ ݔ൅ ݕሻ ௡. The coefficients for the terms in the expansion ofሺ ݔ൅ ݕሻ ௡^ are given in the ݊ ௧௛^ row of Pascal’s Triangle. Here are more: 1 8 28 56 70 56 28 8 1 1 9 26 84 126 126 84 26 9 1 1 10 45 120 210 252 210 120 45 10 1

Synthetic Division and the Binomial Expansion: A Comprehensive Guide, Study notes of Mathematics

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