Binomials increased to a power

A binomial is a polynomial with two terms. We"re going come look in ~ the Binomial expansion Theorem, a shortcut an approach of increasing a binomial to a power.

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(x+y)0 = 1(x+y)1 = x + y(x+y)2 = x2 + 2xy + y2(x+y)3 = x3 + 3x2y + 3xy2 + y3(x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4(x+y)5 = x5 + 5x4y + 10x3y2 +10x2y3 + 5xy4 + y5

There are several things the you hopefully have noticed ~ looking at the expansion

There space n+1 state in the growth of (x+y)n The level of each term is n The strength on x begin with n and decrease come 0 The strength on y start with 0 and increase come n The coefficients room symmetric

Pascal"s Triangle

Pascal"s Triangle, called after the French mathematician Blaise Pascal is one easy way to find thecoefficients that the expansion.

Each heat in the triangle begins and ends with 1. Each element in the triangle is the sum of thetwo elements immediately above it.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 11 7 21 35 35 21 7 1


Combinations will be disputed more fully in ar 7.6, buthere is a brief summary to get you going through the BinomialExpansion Theorem.


A combination is an plan of objects, withoutrepetition, and order no being important. Another definition of combination is the number ofsuch kinds that are possible.

The n and r in the formula stand for the total number of objects to pick from and also the number ofobjects in the arrangement, respectively.

The quickest way, if unimaginative, is to usage the combination function on your calculator. On the TI-82 and TI-83, it is found under the math menu, Probability submenu, choice 3. Go into the value for n first, then the nCr notation, then the worth for r.

Each aspect in Pascal"s Triangle is a mix of n things. The value for r starts with zeroand functions its way up come n. Or, due to the fact that of symmetry, you could say it starts with n and works itsway under to 0.

Let"s think about the n=4 row of the triangle.

4C0 = 1, 4C1 = 4, 4C2 = 6, 4C3 = 4, 4C4 = 1

Notice that the third term is the term with the r=2. That is, we start counting v 0. This willcome right into play later.

Binomial expansion Theorem

Okay, currently we"re prepared to placed it all together.

The Binomial development Theorem have the right to be composed insummation notation, wherein it is really compact andmanageable.


Remember that because the lower limit of the summation begins with 0, the 7th term of the sequenceis actually the term once k=6.

When you go to usage the binomial growth theorem, it"s actually less complicated to put the guidelines indigenous the optimal of this page right into practice. The x starts off to the nth power and goes under by one every time, the y starts turn off to the 0th strength (not there) and also increases through one each time. The coefficients room combinations.

Binomial growth Example:

Expand ( 3x - 2y )5

Start off by figuring the end the coefficients. Remember the these space combinations the 5 things, k in ~ a time, where k is one of two the power on the x or the power on the y (combinations room symmetric, so it doesn"t matter).

C(5,0) = 1; C(5,1) = 5; C(5,2) = 10; C(5,3) = 10; C(5,4) = 5; C(5,5) = 1

Now throw in the 3x and -2y terms.

1(3x)5(-2y)0 + 5(3x)4(-2y)1 + 10(3x)3(-2y)2 + 10(3x)2(-2y)3 + 5(3x)1(-2y)4 + 1(3x)0(-2y)5

Raise the individual components to their ideal powers.

1(243x5)(1) + 5(81x4)(-2y) + 10(27x3)(4y2) + 10(9x2)(-8y3) + 5(3x)(16y4) + 1(1)(-32y5)

Simplify every term to get the final answer.

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243x5 - 810x4y + 1080x3y2 - 720x2y3 + 240xy4 - 32y5

Finding the kth term

Find the ninth term in the development of (x-2y)13

Since we begin counting v 0, the 9th term is in reality going to be when k=8. That is, the strength on the x will 13-8=5 and the strength on the -2y will certainly be 8. The coefficient is one of two C(13,8) or C(13,5), combinations are symmetric, so that doesn"t matter.