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Q: How do you find the coefficients of the terms in the binomial expansion?

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First ("first" terms of each binomial are multiplied together)Outer ("outside" terms are multiplied-that is, the first term of the first binomial and the second term of the second)Inner ("inside" terms are multiplied-second term of the first binomial and first term of the second)Last ("last" terms of each binomial are multiplied)The general form is: (A+B)(C+D) = AC + AD + BC + BDWhere AC is the first, AD is the outer, BC is the inner, and BD is the last.So:(X+4)(X-5)= X^2 - 5X + 4X - 20= X^2 - 1X - 20

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An arithmetic sequence.

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We often come across the algebraic identity (a + b)2 = a2 + 2ab + b2. In expansions of smaller powers of a binomial expressions, it may be easy to actually calculate by working out the actual product. But with higher powers the work becomes very cumbersome.The binomial expansion theorem is a ready made formula to find the expansion of higher powers of a binomial expression.Let ( a + b) be a general binomial expression. The binomial expansion theorem states that if the expression is raised to the power of a positive integer n, then,(a + b)n = nC0an + nC1an-1 b+ nC2an-2 b2+ + nC3an-3 b3+ ………+ nCn-1abn-1+ + nCnbnThe coefficients in each term are called as binomial coefficients and are represented in combination formula. In general the value of the coefficientnCr = n!r!(n-r)!It may be interesting to note that there is a pattern in the binomial expansion, related to the binomial coefficients. The binomial coefficients at the same position from either end are equal. That is,nC0 = nCn nC1 = nCn-1 nC2 = nCn-2 and so on.The advantage of the binomial expansion theorem is any term in between can be figured out without even actually expanding.Since in the binomial expansion the exponent of b is 0 in the first term, the general term, term is defined as the (r+1)th b term and is given by Tr+1 = nCran-rbrThe middle term of a binomial expansion is [(n/2) + 1]th term if n is even. If n is odd, then terewill be two middle terms which are [(n+1)/2]th and [(n+3)/2]th terms.

We can use it to find the coefficients of numbers when we expand a binomial. We also use it in probability theory. In fact there are many uses for it.

To add like terms, find the terms that have the same (or no) variable, and combine the coefficients of the terms. For instance, if you have a+b where a and b are real numbers, you can combine them.

Find the gcf for the coefficients and find the smallest exponential for the variable(s), but the variable must be in all the monomial terms.

The Pascal's triangle is used partly to determine the coefficients of a binomial expression. It is also used to find the number of combinations taken n at a time of m things .

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The key to using the elimination method is to find variable terms in two equations that have unequal coefficients

A binomial is an algebraic expression. It does not have an area.

A polynomial discriminant is defined in terms of the difference in the roots of the polynomial equation. Since a binomial has only one root, there is nothing to take its difference from and so in such a situation, the discriminant is a meaningless concept.

You can find the coefficients of an expanded binomial using the numbers in Pascal's triangle. 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 1 1 7 21 35 35 21 7 1 These are a few rows of Pascals triangle. Now let's look at a few binomials, expanded to the second and third powers. (a+b)2=a2 +2(ab) + b2 notice the coefficients are the numbers in the second row of the triangle above. (a+b)3= a3+3(a2b)+3(ab2)+b3 and once again note that the coefficients are the numberin the third line of Pascal's triangle. The first line, by the way, which is 1,1 is the coefficient of (a+b)1 This will work for any power of the binomial. There are generalized form for non-integer powers.

First ("first" terms of each binomial are multiplied together)Outer ("outside" terms are multiplied-that is, the first term of the first binomial and the second term of the second)Inner ("inside" terms are multiplied-second term of the first binomial and first term of the second)Last ("last" terms of each binomial are multiplied)The general form is: (A+B)(C+D) = AC + AD + BC + BDWhere AC is the first, AD is the outer, BC is the inner, and BD is the last.So:(X+4)(X-5)= X^2 - 5X + 4X - 20= X^2 - 1X - 20

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