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Chapter 14: Problem 27

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(3 x-y)^{5}$$

Short Answer

Expert verified
The expanded form of the binomial is \(243x^5 - 1215x^4*y + 4050x^3*y^2 - 8100x^2*y^3 + 6075x*y^4 - y^5\)

Step by step solution

01

Define the binomial theorem

The binomial theorem states that for any numbers a and b, and for any natural number n, \((a + b)^n = 鈭慱{k=0}^{n} (nCk)*(a^{n-k})*(b^k)\). nCk represents the number of combinations of n items taken k at a time.
02

Identify a, b and n

In the given problem, the equation is \((3x - y)^5\) thus a is 3x, b is -y and n is 5.
03

Substitute a, b and n values into formula

Applying these values to the binomial theorem equation, we use each term in the series for k=0 through n, replacing n, a and b where necessary and simplifying each term. It will look like this: \((3x - y)^5 = 鈭慱{k=0}^{5} (5Ck)*(3x^{5-k})*(-y^k)\)
04

Expand the binomial

The expanded form of the equation will thus be: \((3x - y)^5 = 243x^5 - 5*243x^4*y + 10*81x^3*y^2 - 10*27x^2*y^3 + 5*9x*y^4 - y^5\)

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{4} 3 i+\sum_{i=1}^{4} 4 i=\sum_{i=1}^{4} 7 i$$

Use the formula for the value of an annuity to solve Exercises. Round answers to the nearest dollar. To offer scholarship funds to children of employees, a company invests \(\$ 15,000\) at the end of every three months in an annuity that pays \(9 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of ten years? b. Find the interest.

Find the term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) containing \(x^{4}\) as a factor.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\). [Hint: Write \(x^{2}+x+1\) as \(\left.x^{2}+(x+1) .\right]\)

Rationalize the denominator: \(\frac{6}{\sqrt{3}-\sqrt{5}}\).
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