Algebra

Sequences and Series media type="custom" key="6795913"

Infinite Geometric Series media type="custom" key="6795919"

Exponents and Logarithms media type="custom" key="6795921"

Binomial Expansion media type="custom" key="6810441"

To determine the coefficient of the terms in a binomial expansion, math (x+a)^n math we could either use Pascal's Triangle or the Binomial Formula. Let us illustrate the use of these two methods by expanding math (x-2y)^4 math

1. Pascal's Triangle We can make use of Pascal's Triangle to determine the coefficient of the terms in the expansion. Note that for this case, n=4 therefore we shall be looking at the 5th level of Pascal's Triangle media type="custom" key="6810907" The binomial coefficients from are 1,4,6,4,1. Therefore the expansion is as follows. math (x-2y)^4 =(1)x^4-(4)x^3(2y)+(6)x^2(2y)^2-(4)x(2y)^3+(1)(2y)^4 \\ = x^4-8x^3y+24x^2y^2-32xy^3+16y^4 math

2. Using the Binomial Formula The Binomial Formula is given by math (x+a)^n = _nC_0x^n+_nC_1x^{n-1}a+_nC_2x^{n-2}a^{2}+\ldots+_nC_rx^{n-r}a^r+\ldots+a^n \\ & = \dbinom {n}{0} x^n + \dbinom{n}{1} x^{n-1}a + \dbinom{n}{2}x^{n-2}a^2+ \ldots + \dbinom{n}{r} x^{n-r}a^r+\ldots+a^n\\ math where math _nC_r=\dbinom{n}{r}=\displaystyle \frac{n!}{r!(n-r)!} math Going back to the expansion, we have n=4 then math (x-2y)^4=\dbinom{4}{0}x^4+\dbinom{4}{1}x^{4-1}(-2y)+\dbinom{4}{2}x^{4-2}(-2y)^2+\dbinom{4}{3}x^{4-3}(-2y)^3+\dbinom{4}{4}x^{4-4}(-2y)^4 \\ & = \dfrac{4!}{0!(4-0)!}x^4+\dfrac{4!}{1!(4-1)!}x^3(-2y)+\dfrac{4!}{2!(4-2)!}x^2(-2y)^2+\dfrac{4!}{3!(4-3)!}x(-2y)^3+\dfrac{4!}{4!(4-4)!}(-2y)^4\\ & = x^4-8x^3y+24x^2y^2-32xy^3+16y^4 math Note the following: math 0!=1\\ 1!=1 math

If for instance, the problem asks us for the coefficient of the given term; this can still be done without having to actually expand the binomial and going through each of the terms. For this case, we shall use the formula for the general term which is math t_{r+1}=\dbinom{n}{r}x^{n-r}a^{r} math

Example: Find the terms indicated in the expansions of the following expressions. math 1.(x+y)^7 term:x^5y^2 math

For the example above a=y and since the exponent of y is 2 then using the formula for the general term, r=2 then math t_{2+1}=\dbinom{7}{2}x^{7-2}y^2\\ t_3=\dfrac{7!}{2!(7-2)!}x^5y^2\\ & = \dfrac{7!}{2!5!}x^5y^2 & = 21x^5y^2 math

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