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Use the binomial Theroum to expand the expression and express the results in simplified form (3x...

Question:

Use the binomial Theroum to expand the expression and express the results in simplified form {eq}(3x + 2)^3. {/eq}

Binomial Theorem:

If an expression is given into the following form:

{eq}(a + b)^n {/eq}, where n is any positive integer.

Then, the given expression can be expanded by using the binomial theorem. The following binomial expansion is:

{eq}(a + b)^n =\, ^nC_0 a^n b^0 + \,^nC_1 a^{n-1} b^1 + \, ^nC_2 a^{n-2} b^2 + ... + \,^nC_n a^0 b^n {/eq}

Answer and Explanation:

The given expression is:

{eq}(3x + 2)^3 {/eq}

We have:

{eq}a = 3x, \quad b = 2, \quad n = 3 {/eq}

Using the binomial theorem, we get:

{eq}(3x + 2)^3 = \, ^3C_0 ((3x)^3) (2^0) + \,^3C_1 ((3x)^{3-1}) (2^1) + \, ^3C_2 ((3x)^{3-2}) (2^2) + \,^3C_3 ((3x)^0) (2^3) \\ (3x + 2)^3 = 1(27x^3)(1) + 3(9x^2)(2) + 3(3x)(4) + 1(1)(8) \\ \boxed{(3x + 2)^3 = 27x^3 + 54x^2 + 36x + 8} {/eq}

This is the binomial expansion of the given expression.


Learn more about this topic:

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The Binomial Theorem: Defining Expressions

from Algebra II: High School

Chapter 12 / Lesson 7
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