Q. 311 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. 311 (page 1220)

In the following exercise, expand each binomial, using the Binomial Theorem.
${(t - 1)}^{9}$

Short Answer

Expert verified

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

$(t - 1)^{9} = t^{9} - 9 t^{8} + 36 t^{7} - 84 t^{6} + 126 t^{5} - 126 t^{4} + 84 t^{3} - 36 t^{2} + 9 t - 1$

Step by step solution

Step 1. Given information

The given expression is ${(t - 1)}^{9}$

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 ${(t - 1)}^{9}$ , we have:

$a = t b = - 1 n = 9$

Using the binomial expansion, we get:

$(t - 1)^{9} = (\begin{array}{l} 9 \\ 0 \end{array}) t^{5} + (\begin{array}{l} 9 \\ 1 \end{array}) t^{9 - 1} (- 1)^{1} + (\begin{array}{l} 9 \\ 2 \end{array}) t^{9 - 2} (- 1)^{2} + (\begin{array}{l} 9 \\ 3 \end{array}) t^{9 - 3} (- 1)^{3} + (\begin{array}{l} 9 \\ 4 \end{array}) t^{9 - 4} (- 1)^{4} + (\begin{array}{l} 9 \\ 5 \end{array}) t^{9 - 5} (- 1)^{5} + (\begin{array}{l} 9 \\ 6 \end{array}) t^{9 - 6} (- 1)^{6} + (\begin{array}{l} 9 \\ 7 \end{array}) t^{9 - 7} (- 1)^{7} + (\begin{array}{l} 9 \\ 8 \end{array}) t^{9 - 8} (- 1)^{8} + (\begin{array}{l} 9 \\ 9 \end{array}) (- 1)^{9}$

Step 3. Simplifying the expansion

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

(t - 1)^{9} = (1) t^{5} + (9) t^{8} (- 1) + \frac{9!}{2! 7!} t^{7} (1) + \frac{9!}{3! 6!} t^{6} (- 1) + \frac{9!}{4! 5!} t^{5} (- 1) + \frac{9!}{4! 5!} t^{4} (- 1) + \frac{9!}{3! 6!} t^{3} (1) + \frac{9!}{7! 2!} t^{2} (- 1) + \frac{9!}{1! 8!} t^{1} (1) + (1) (- 1) (t - 1)^{9} = t^{9} - 9 t^{8} + 36 t^{7} - 84 t^{6} + 126 t^{5} - 126 t^{4} + 84 t^{3} - 36 t^{2} + 9 t - 1

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91影视

In the following exercise, expand each binomial, using the Binomial Theorem.
${(t - 1)}^{9}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the expression

Step 3. Simplifying the expansion

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91影视

In the following exercise, expand each binomial, using the Binomial Theorem.t-19

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the expression

Step 3. Simplifying the expansion

One App. One Place for Learning.

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In the following exercise, expand each binomial, using the Binomial Theorem.
${(t - 1)}^{9}$