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Chapter 11: Problem 34

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x-2 y)^{9} $$

Short Answer

Expert verified
The first three terms of the binomial expansion are \(x^9\), -18x^8y and 144x^7y^2.

Step by step solution

01

Define the coefficients, variables and powers

Identify the values of a, b and n in the problem. Here, a = x, b = -2y and n = 9. The coefficients for the first three terms can be found by using the binomial coefficient \[\binom{n}{k}\] which is equal to n!/(k!(n-k)!). Here, n is 9. For the first term k=0, second term k=1 and for the third term k=2.
02

Calculate the first term

The first term is obtained when k=0 in the binomial coefficient, which simplifies to \[\binom{9}{0}\] and equals 1. Since a = x and n = 9 the first term becomes \[1*x^9*(-2y)^0\] which simplify to \(x^9\).
03

Calculate the second term

The second term is obtained when k = 1. So, the binomial coefficient becomes \[\binom{9}{1}\] which equals 9. Thus the second term becomes \[9*x^8*(-2y)^1\] which simplifies to -18x^8y.
04

Calculate the third term

The third term is obtained when k = 2. The binomial coefficient \[\binom{9}{2}\] equals 36. The third term becomes \[36*x^7*(-2y)^2\], which simplifies to 144x^7y^2.

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Most popular questions from this chapter

Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) show that $$1+2+3+\cdots+n=\frac{n(n+1)}{2}$$ is true for the given value of \(n\) $$ n=5: \text { Show that } 1+2+3+4+5=\frac{5(5+1)}{2} $$

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). A pendulum swings through an arc of 16 inches. On each successive swing, the length of the arc is \(96 \%\) of the previous length. $$ \begin{array}{cccc} {16,} & {0.96(16),} & {(0.96)^{2}(16),} & {(0.96)^{3}(16)} \\ {\text { Ist }} & {2 \text { nd }} & {3 \text { rd }} & {4 \text { th }} \\ {\text { swing }} & {\text { swing }} & {\text { swing }} & {\text { swing }} \end{array} $$ After 10 swings, what is the total length of the distance the pendulum has swung?

Graph \(f(x)=x^{2} .\) Then use the graph of \(f\) to obtain the graph of of \(g(x)=(x+2)^{2}-1\)

Explaining the Concepts Give an example of an event whose probability must be determined empirically rather than theoretically.

Convert the equation $$ 4 x^{2}+y^{2}-24 x+6 y+9=0 $$ to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of the foci. (Section \(10.1,\) Example 5 )
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