Q. 312 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. 312 (page 1220)

In the following exercise, expand each binomial, using the Binomial Theorem.
${(2 x + 1)}^{4}$

Short Answer

Expert verified

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

$(2 x + 1)^{4} = 16 x^{4} + 32 x^{3} + 24 x^{2} + 8 x + 1$

Step by step solution

Step 1. Given information

The given expression is ${(2 x + 1)}^{4}$

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 ${(2 x + 1)}^{4}$ , we have:

$a = 2 x b = 1 n = 4$

Using the binomial expansion, we get:

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

Step 3. Simplifying the expansion

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

$(2 x + 1)^{4} = (1) (2 x)^{4} + (4) (2 x)^{3} (1) + \frac{4!}{(4 - 2)! 2!} (2 x)^{2} (1) + \frac{4!}{(4 - 3)! 3!} (2 x)^{'} (1) + (1) (1) (2 x + 1)^{4} = 16 x^{4} + 32 x^{3} + 24 x^{2} + 8 x + 1$

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91影视

In the following exercise, expand each binomial, using the Binomial Theorem.
${(2 x + 1)}^{4}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the expression

Step 3. Simplifying the expansion

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Step by step solution

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In the following exercise, expand each binomial, using the Binomial Theorem.
${(2 x + 1)}^{4}$