Q. 314 In the following exercise, expan... [FREE SOLUTION]

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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 12: Q. 314 (page 1220)

In the following exercise, expand each binomial, using the Binomial Theorem.
${(x - 3 y)}^{5}$ .

Short Answer

Expert verified

The expansion of the given expression by binomial theorem is given as:

$(x - 3 y)^{5} = x^{5} - 15 x^{4} y + 90 x^{3} y^{2} - 270 x^{2} y^{3} + 405 x y^{4} - 243 y^{5}$

Step by step solution

Step 1. Given information

The given expression is ${(x - 3 y)}^{5}$

Step 2. Expanding the expression

The binomial expansion of the expression ${(a + b)}^{n}$ is given as:

$(a + b)^{n} = (\begin{array}{l} n \\ 0 \end{array}) a^{n} + (\begin{array}{l} n \\ 1 \end{array}) a^{n - 1} b^{1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{n - 2} b^{2} + 鈥� + (\begin{array}{l} n \\ r \end{array}) a^{n - r} b^{r} + 鈥� + (\begin{array}{l} n \\ n \end{array}) b^{n}$

Similarly, for ${(x - 3 y)}^{5}$ , we have:

$a = x b = - 3 y n = 5$

Using the binomial expansion, we get:

$(x - 3 y)^{5} = (\begin{array}{l} 5 \\ 0 \end{array}) (x)^{5} + (\begin{array}{l} 5 \\ 1 \end{array}) (x)^{5 - 1} (- 3 y)^{1} + (\begin{array}{l} 5 \\ 2 \end{array}) (x)^{5 - 2} (- 3 y)^{2} + (\begin{array}{l} 5 \\ 3 \end{array}) (x)^{5 - 3} (- 3 y)^{3} + (\begin{array}{l} 5 \\ 4 \end{array}) (x)^{5 - 4} (- 3 y)^{4} + (\begin{array}{l} 5 \\ 5 \end{array}) (- 3 y)^{5}$

Step 3. Simplifying the expansion

Simplifying the exponents and evaluating the binomial coefficient, we get:

$(x - 3 y)^{5} = (1) (x)^{5} + (5) (x)^{4} (- 3 y)^{1} + \frac{5!}{(5 - 2)! 2!} (x)^{3} (9 y^{2}) + \frac{5!}{(5 - 3)! 3!} (x)^{2} (- 27 y^{3}) + \frac{5!}{(5 - 4)! 4!} (x)^{1} (81 y^{4}) + (1) (- 243 y^{5}) (x - 3 y)^{5} = x^{5} - 15 x^{4} y + 90 x^{3} y^{2} - 270 x^{2} y^{3} + 405 x y^{4} - 243 y^{5}$

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91影视

In the following exercise, expand each binomial, using the Binomial Theorem.
${(x - 3 y)}^{5}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the expression

Step 3. Simplifying the expansion

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In the following exercise, expand each binomial, using the Binomial Theorem. x-3y5.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the expression

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In the following exercise, expand each binomial, using the Binomial Theorem.
${(x - 3 y)}^{5}$ .